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MATHEMATICAL CONSTRUCTION 

INFORMAL NUMBER WORK 
FOR BUSY HANDS 

GRADES ONE AND TWO 



BY 

N. LOUISE LAFFIN 

OF THE CHICAGO SCHOOLS 



A. FLANAGAN COMPANY 

CHICAGO 






Copyright 1911 

BY 

N. LOUISE LAFFIN 



$ C 



©CI.A303189 



TABLE OF CONTENTS 

Introduction 5 

Plans of Lessons 12 

All the combinations to 12, and the lessons in which 

they are found 16 

FOLDING. 
LESSON. PAGE. 

I Shawl 23 

II Booklet 25 

III Wall-Pocket 26 

IV Soldier's Cap 28 

V Fireman's Cap 29 

VI Envelope 30 

VII Picture Frame 32 

VIII Salt and Pepper Holder 35 

IX Boat 36 

X Square Prism 37 

XI Cube 40 

XII Cradle 42 

XIII Chair 43 

XIV Buggy 45 

XV to XXV Baskets 46 

FOLDING AND WEAVING. 

I Making 2 cuts on mat (16 squares), 

weaving with one strip 53 



CONTENTS 
LESSON. PAGE. 

II Making 2 cuts on mat (12 squares), 

weaving with one strip 53 

III Making 3 cuts on mat, weaving with 2 

equal strips 54 

IV Making 3 cuts on mat, weaving with 4 

equal strips 54 

V Making 5 cuts on mat, weaving with 2 

equal strips 55 

VI Making 5 cuts on mat, weaving with 4 

equal strips &> 

VII Making 7 cuts on mat, weaving with 6 

equal strips 56 

VIII Making 5 cuts (relation 1 to 2), weaving 

with 4 strips (relation 1 to 2) 56 

IX Making 5 cuts (relation 1 to 2), weaving 

with 5 strips (relation 1 to 2) 57 

X Making 4 cuts (relation 1 to 4), weaving 

with 3 strips (relation 1 to 4) 57 

XI Making 7 cuts (relation 1 to 2), weaving 
with 7 strips (relations 1 to 2 and 1 

to 4) over 2 and under 1 58 

XII Relation 1 to 2 cutting mat, 1 to 4 cut- 
ting strips, weaving over 3 and under 1. 58 

FREE CUTTING AND WEAVING. 

I Relation 1 to 2 60 

II Relation 1 to 4 61 

III Relation 1 to 3 (4 cuts) 61 

IV Relation 1 to 3 (6 cuts) 62 



CONTENTS 
LESSON. PAGE. 

V Eelation 1 to 3 (8 cuts) 62 

VI Relation 1 to 3 (7 cuts) 63 

VII Eelation 1 to 5, 1 to 2 (6 cuts) 63 

WALL-PAPER MADE BY FREE-CUTTING 65 

PAPER RINGS MADE BY EREE-CUTTING 67 

MEASURING AND WEAVING. 

I Bank Decoration. (Rulers: I", 2", 3", 

4") 68 

II Calendar Back. (Rulers: 1" ', 2", 3". 4", 

7") 69 

III Blotter. (Rulers: 6", 5", 4", 3", 2", 

1") 70 

IV Napkin Ring. (Rulers: 1", 2", 3", 4", 

5", 6") 72 

V Telephone Pad. (Rulers : 1", 4", 5", 8") 73 
VI Wall-pocket for Letters. (Rulers: 1", 2", 

3", 7", 10") 74 

VII Circular Woven Basket. (Rulers: 1", 2", 

5", 6", 9", 10") 76 

VIII Needle-book. (Rulers: 1", 3", 4", 7", 

8", 10") 78 

PROGRESSIVE MEASURING LESSONS. 

Introduction of 
Single Unit Rulers. 

I Scissors Holder. (Ruler : 8") 80 

II Mayflower. (Rulers : 8", 4") 81 

III Cup. (Rulers : 6", 3") 83 



CONTENTS 
LESSON. PAGE. 

IV Book-mark. (Rulers : 4", 2") 85 

V Stamp Pocket. (Rulers : 2", 3", 5") . . . 86 
VI Washcloth Pocket. (Rulers: 1", 7", 3", 

4") 88 

VII Pilgrim's Bonnet. (Rulers: 2", 4" ', 6") 89 

VIII Stove (Rulers : 3", 6", 9") 91 

IX Bank. (Rulers : 4", 8", 12") 93 

X Envelope (for decoration units) (Rulers: 

5", 1", 7", 2") 95 

XI Taboret. (Rulers: 5", 3", 8", 1") 96 

XII Father Bear's Chair. (Rulers: 3', 6", 

9", 12") 97 

XIII Mother's Chair. (Rulers: 2", 4", 6", 

8") 99 

XIV Baby's Chair. (Rulers: 1", 2", 3", 4"). 100 
XV Father's Bed. (Rulers: 3", 6", 9", 12"). 101 

XVI Mother's Bed. (Rulers: 2", 4", 6", 8"). 103 

XVII Baby's Bed. (Rulers: 1", 2", 3", 4") . . 103 
XVIII Father Bear's Bowl. (Rulers: 12", 4", 

3", 1") 103 

XIX Mother's Bowl. (Rulers: 12", 3", 2", 

1") 104 

XX Baby's Bowl. (Rulers: 12", 2", 1")... 105 

XXI Jack's Pail. (Rulers: 9", 5", 4", 1") . . 106 

XXII Jill's Pail. (Rulers: 9", 3", 2", 1").. 107 

XXIII Fox's Dish. (Rulers: 5", 2", 1") 108 

XXIV Stork's Dish. (Rulers: 5", 7", 6", 1"). 109 
XXV Handbag. (Rulers: 9", 6", 3", 2", 1"). 109 

XXVI Pencil-Box With Lid. (Rulers: 10", 2", 

3", 5") Ill 



CONTENTS 
LESSON. PAGE. 

XXVII Match-Safe. (Eulers: 5", 4", 3", 2", 

1") 112 

XXVIII Wood-Box. (Rulers: 
3", 6", 9", 12" 

2", 4", 6", 8") 113 

XXIX Sled. (Rulers: 5", 4", 3", 2", 1") .... 114 

XXX Pushcart. (Rulers: 7", 5", 2", 1") 115 

XXXI Gocart. (Rulers: 2", 4", 6", 8") 117 

XXXII Cradle. (Rulers : V, 9", 5", 4") 119 

XXXIII Bureau. (Rulers: 3", 6", 9", 12" 

8", 2", 5") 121 

XXXIV Chiffonier. (Rulers: 2", 4", 6", 8", 

10") 123 

XXXV A Large Envelope. (Rulers: 11", 1", 

8", 4") 125 

XXXVI Book for Cuttings — unfolded sheets — 

(Rulers : 1", 3", 6", 9") 127 

XXXVII Book for Words— unfolded sheets— (Rul- 
ers: 6", 1", 7") 128 



MATHEMATICAL CONSTKUCTION 



INTRODUCTION 

All the construction work should be carefully selected by 
the teacher with, a two-fold purpose. 

1st. The appropriateness of the thing itself to the child's 
need; a co-relation to other studies, or a gift, plaything or 
actual necessity. 

2nd. The appropriateness of the number relations to the 
child's mathematical discoveries; a proper sequence from the 
simplest of mathematical relations to the more complex, and 
a proper order of inferences. 

The possible number relations found in the process of 
the construction of an article should be brought out carefully 
by the teacher. While the child is making something he is 
in a receptive mood for this number work. Each step gives 
new surfaces or lines to compare. 

When the teacher artfully questions, giving concrete 
problems regarding these ever changing lines and surfaces, 
she gets answers from alert minds ; minds that are comparing 
and coming to conclusions then and there; minds that are 
not trying to remember what the teacher said yesterday or a 
week ago; minds that are seeing and drawing inferences at 
the present moment. 

In applying number work to the making of things in the 

5 



q MATHEMATICAL CONSTEUCTION 

first and second grades (by folding, free cutting or measur- 
ing), a teacher should let the child judge the relations of 
things himself. By repeated acts of judgment he gains the 
power to see relations correctly. 

Indefinite relations are seen before the definite. There- 
fore, the earliest lessons (found in the folding of articles) 
should deal only with words which express contrast; such 
as larger and smaller, longer and shorter, more or less. A few 
weeks of this indefinite relation work usually suffice. Then 
follow the relations, 1, 2, 3, 4, equal -J, J, J. In these 
lessons the teacher should only show the fold, the child 
imitating. This gives mental training through sight. After 
the child has imitated the teacher's silent direction, the 
teacher should ask questions bringing out the relations 
(indefinite or definite as the state of the child's mind 
permits). 

These questions should introduce technical words inci- 
dentally and informally to the pupil. By giving little concrete 
problems based on the relation of surfaces, lines and solids, 
she can do this. For instance: when a square is folded into 
equal oblongs she may say, "If this oblong is enough cloth 
(pointing to one oblong) for one doll's dress, what is this 
oblong?" (pointing to other). Or, "Play this oblong is a 
dresser scarf, which has lace all around it. If the lace on 
the upper edge costs a dime, what does the lace on the side 
edge cost?" Ans. "More than a dime." Child gets words 
oblong, upper edge, lower edge incidentally. 

Giving concrete problems in this way is interesting to 
the imaginative mind of a child. He plays that sand is 
sugar, buttons are money, and stones are potatoes or apples. 
Why can't he play that oblongs or squares are parks, play- 



INTRODUCTION 7 

grounds, ceilings, bed quilts, tablecloths, articles of clothing, 
etc.? 

And why can't he also play that two of those are two 
parks, or three of them are three parks? If one park con- 
tains five acres, two parks contain two five acres, etc. Why 
can't he play that a triangle is : a doll's shawl, a pond of ice 
to skate on, a tile, a piece of glass, the leather corner of 
a desk blotter, etc.? Why can't he play that edges require 
fringe for rugs, bedspreads, napkins, doilies, the top of a 
carriage, etc.; fences for parks, farms, yards, empty lots, 
etc. ; binding for slates, books, pictures, cloth ; curbstones for 
streets ; hedges or trees in a row for yards, parks, roads ? 

When the children are familiar with the technical words, 
the teacher should give verbal directions instead of silently 
showing them, thus giving child mental training through 
hearing. These verbal directions are more difficult than the 
sight directions, for they involve a memory of the technical 
words and number relations learned incidentally through 
the concrete problem work. 

After a child can fold, by following the verbal directions 
of the teacher, he is ready to use a ruler. An ordinary 
12-inch ruler is beyond the comprehension of the beginner. 
The inch is within the 2-inch measure, and within the 3-inch 
unit there is the 2-inch and also the 1-inch unit. The child 
cannot see the relation of 1 inch to 2 inches on the ruler; 
therefore, he should use single unit rulers. A teacher should 
have a complete set for each child in the room. A set consists 
of 12 pieces of strong, tough, smooth cardboard (pressboard 
is the proper name). Each piece is 1 inch wide, and 
respectively, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 inches 
long. 



MATHEMATICAL CONSTRUCTION 



UD 



Of course the child is not given the whole set at once. 
He would be confused. 

If the teacher intends making something which requires 
a 6-inch square, for instance, she gives him a 6-inch ruler. 
If that must be bisected he is given a 3-inch ruler. If the 
original paper is 9"xl2" let him estimate the width of it. 
He will tell you that the width is 

the sum of 3" and 6" or 

the sum of 6" and i/ 2 of 6" or 

three 3" 
He will say that the length is : 

four 3" or 

the sum of 6" and 6" or 

the sum of 6" and 3" and 3" or 

the sum of three 3" and l/ 2 of 6" 



INTRODUCTION 9 

With these rulers one obtains mathematical relations of 
the measures which one could never get from little children 
with the ordinary ruler, which has the units within many 
other units. The children see: 

the sum of 3" and 3", 
the difference between 6" and 3", 
the division of 6"x2, 
the multiplication of 3"x2. 
When in a later lesson the 9-inch ruler is introduced, the 
children see: 

6" 3'3"=9" $ of 9"=3" 



3 



// 



9" 




9" 


9' 


-6" 


-3 



3" 6" 

In making an article based on a square consisting of 16 
squares, we use the dimensions : 

3, 6, 9, 12, or - 

2, 4, 6, 8, or 

1, 2, 3, 4. 

Why should a child be bothered with all the units within 
units of an ordinary ruler when he can be given something 
which is definite to him? How easy it is to see that 6 is J 
of 12; 3 is J of it, etc., with these distinct units of measure. 

The making of an article based on a square consisting 
of 9 squares vividly shows the relation of 1, 2, 3; 2, 4, 6; 
3, 6, 9; 4, 8, 12. " 



10 MATHEMATICAL CONSTEUCTION 

One based on a square consisting of 4 squares gives the 
relations 1, 2 ; 2, 4 ; 3, 6 ; 4, 8 ; 5, 10 ; 6, 12. 

The first step in the using of the ruler is like the first step 
in folding. The teacher should show the direction and the 
child imitate (sight and mind training). The only verbal 
expression of the teacher is the naming of the ruler, viz. : 
"This is a 6-inch ruler." 

The second point in the use of the ruler is that the teacher 
directs verbally, after the child has learned how to use it. 

Third point: Child must do mental addition, subtrac- 
tion, division, or multiplication, by following teacher's verbal 
direction. 

Teacher asks questions like these : 

(Suppose child to have rulers 3", 6", 9", 12" long, and 
that he is familiar with their relations through former work 
in measuring, and that he has paper 9"xl2".) 

Show me the edge that is three 3" long. 

How many inches long is it ? 

Show me the edge that is the sum of 9" and 3" long. 

How many inches long is it? 

On the upper long edge mark off J of 12", One-half of 
12" is how many inches? 

If it costs $12.00 to build a fence on this edge (pointing 
to the 12-inch edge) what part of $12.00 will it cost to build 
a fence on this part (showing 6") of it? Ans. -J of $12.00. 

How much money will it cost? ($6.00) Etc. 

This last question relative to cost is an inference. From 
actually seeing that 6" are -| of 12" the child easily infers 
that $6.00 are -J of $12.00. If questions were asked concern- 
ing time required to build the fence it would be easy for the 
child to infer that 6 days are J of 12 days ; 



INTRODUCTION n 



6 hours are -| of 12 hours; 



6 men are -J of 12 men. 

So all possible combinations, separations, products and 
parts of the numbers to 12 can be reached through the use 
of these rulers. 

When a child has seen that the sum of 3"+3"=6" and has 
inferred that 

3c and 3c = 6c ; or 

3 yds. and 3 yds. = 6 yds. ; 

3 hours and 3 hours = 6 hours ; 
the teacher should go to the board and write 3 

3 



At some other time, other than the construction period, 
have a drill on writing the combination. Write it as a whole. 
Erase and let them write it. Write it again omitting one of 
the numbers. Erase. Let children write it inserting the one 
the teacher omitted, etc. 

Drills of this sort help to fix number relations which 
have been previously seen. 



MATHEMATICAL CONSTKUCTION 

PLAN OF THE LESSONS 
Folding 

I have in the folding lessons shown what variety of con- 
crete problems can be given; how many of them can be 
correlated to the finished article; how they do incidentally 
show the number relations 1, 2, 3, 4 and more, equal, §, J, J. 

At the end of some of the simple folding lessons, I have 
drawn surfaces which the child sees in making the article. 
These can be put on the blackboard and concrete problems 
based on them after the folding lesson. 

Folding and Free Cutting 
(Weaving and Decorating) 

After a child has dealt with the relations 1, 2, 3, 4, equal, 
•J, J, J in folding lessons, he is ready to do some free cutting. 

Weaving and Decorating give many opportunities for 
cutting a piece of paper into 2 equal pieces, or 3, or 4, as the 
case may be; or, for cutting a piece of paper twice as large, 
long, or wide as another. 

There is a natural sequence of the three topics folding, 
free cutting and measuring in the single subject Weaving. 
The weaving lessons are presented in this order. No concrete 

12 



PLAN OF THE LESSONS 13 

problems have been written in connection with these lessons, 
but they should be given wherever there are surfaces or lines 
to compare. 

These woven mats can be utilized in many ways. They 
can be used in the doll house for mats, table cloths, dresser 
scarfs, sofa-pillow tops; they make a good background for 
wall-pockets, calendars, match scratchers; woven a certain 
way they make telephone pads; they also make a pretty top 
for blotters, or outer side of napkin rings when they are 
under transparent celluloid. Wall paper can sometimes be 
made by weaving. Woven mats can be folded and pasted 
into cornucopias, lanterns, and all sorts of baskets. They 
can be used to decorate books, outside of banks, envelopes, etc. 

In the folding and free cutting weaving lessons, the 
development of the relations 1, 2, 3, 4, has been shown. 

In the measuring lessons, besides showing how naturally 
and definitely number relations are felt in using these single 
unit rulers, a utility has been given for every mat. For 
progressive number work, however, carefully read the lessons 
given under the heading "Measuring with Single Unit 
Rulers/' 

Measuring with Single Unit Kulers 

In measuring with the Single Unit Rulers, I have shown 
how, through a series of lessons, the child finds all combina- 
tions, separations, products, and parts of numbers up to 12; 
also the great variety of lines, surfaces and solids there are 
to compare; how I correlate my concrete problems with the 
finished article. 

There is such a great variety of these surfaces and lines 
that the teacher must discriminate in comparing them. She 



14 



MATHEMATICAL CONSTKUCTION 



must not ask questions on every possible relation she sees; 
if she stops too often the child will lose interest in what he is 
making. That is worse than losing the number work. 

In making an article based on a square consisting of 4 
small squares, the child sees the relation of 

1 to 2; 2 to 4; 3 to 6; 4 to 8; 5 to 10; or 6 to 12. 



"5 






/2. 



<0 ■ 

xj : 

\ 









PLAN OF THE LESSONS 15 

From these squares the following articles can be made: 
Booklet. Picture frame. 

Wall-pocket. Salt and peppers. 

Soldier's cap. Boat. 

Bookmark. Basket. 

Fireman's cap. Cup. 

Scissors holder. 

In making an article based on a square consisting of 9 
small squares, the child sees the relations of 1, 2, 3; 2, 4, 6; 
3, 6, 9; or 4, 8, 12. From 9 squares the following can be 
made: 

Taboret. Dutch bonnet. 

Stove. Wheel-barrow. 

Cube bank. Box. 

Baskets (cut on straight folds or cut on diagonals). 

In making an article based on a square consisting of 16 
small squares, the child observes the relations of 1, 2, 3, 4; 
2, 4, 6, 8 ; or 3, 6, 9, 12. From 16 squares the following can 
be made: 

Baskets (cut on straight fold, or cut on diagonal). 



Cradle. 


Hallowe'en lantern. 


Table. 


Pilgrim cradle. 


Chair. 


Square prism. 


Lounge. 


Square pyramid. 


Bed. 


Cube. 


House. 


Wagons and carts of all kinds. 


Wood box. 


Sled. 


Box with lid. 


Pocket-book. 



16 MATHEMATICAL CONSTEUCTION 

Dresser, sectional book case, or any kind of cabinet with 
drawers or compartments. 

Then there are all sorts of odd lengths and widths of 
oblongs to be used as sides for circular baskets, pails, bowls; 
backgrounds for calendars, match scratchers, wall-pockets; 
strips and mats for weaving. These give such combinations 
as: 

2 3 5 9 7 

3 4 4 2 3 



5 7 9 11 10 etc. 

The following chart shows every combination of numbers 
from 1 to 12 inclusive, and the lessons in which the child 
sees these combinations : 

All the Combinations to 12 and the Lessons in Which 
They are Found 

Note. — These are in the measuring section unless otherwise specified. 

1 Lessons : XIV, XVII, XX, XXII, XXVII. 

1 



2 Lessons: XIV, XVII, XIX, XXII, XXV, XXVII, 
1 XXIX. 



PLAN OF THE LESSONS 17 

3 Lessons : VI, XIV, XVII, XVIII, XXVII, VIII, 
1 (Weaving, p. 78), II (Weaving, p. 69), XXIX. 



4 Lessons: XXX, XXI, XXVII, XXIX. 

1 



5 Lessons: XXX, X, XXIV, VII (Weaving, p. 76). 
1 



6 



6 Lessons: XXIV, XXXVII. 
1 



7 Lessons: II (Weaving, p. 69), VIII (Weaving, p. 78) 
1 



8 



8 Lesson: XXXV. 
1 



9 



18 MATHEMATICAL CONSTKUCTION 

9 Lessons: XXXII, VII (Weaving, p. 76). 
1 

10 

10 Lesson: VII (Weaving, p. 76). 
1 

11 

11 Lesson: XXXV. 
1 

12 

2 Lessons: IV, XIII, XVI, XXVII, XXXI, VII, XIV, 
2 XVII, XXVIII, XXXIV. 



3 Lessons: V, XXVI, XXVII, XXIX. 

2 



4 Lessons: XXX, VII, XIII, XVI, XXVIII, XXXI, 
2 XXXIV, VI (Weaving, p. 74). 



5 Lessons: XXX, X, XXXIII. 

2 



PLAN OF THE LESSONS 19 

6 Lessons: XIII, XVI, XXVIII, XXXI, XXXIV. 
2 

8 

7 Lesson : XXX. 
2 



8 Lessons: XXXI, XXXIII, XXXIV, VI (Weaving, p. 
2 74). 



10 

9 Lesson: VII (Weaving, p. 76). 
2 

11 

10 Lessons : XXXIV, XXXV. 
2 

12 

3 Lessons : III, VIII, XII, XXVIII, XXXIII, XXXVI. 
3 

6 

4 Lessons: VI, VIII (Weaving, p. 78), II (Weaving p. 
3 69). 



20 MATHEMATICAL CONSTEUCTION 

5 Lesson : XI. 
3 

8 

6 Lessons: VIII, XII, XV, XXV, XXVIII, XXXIII, 
3 XXXVI. 

9 

7 Lessons: VI (Weaving, p. 74), VIII (Weaving, p. 
3 78). 

10 

8 Lesson: VIII (Weaving p. 78). 
3 



11 



9 Lessons: XII, XV, XXVIII, XXXIII. 
3 



12 



4 Lessons: IX, XIII, XVI, XXVIII, XXXI, XXXIV, 

4 XXXV. 

8 

5 Lessons : XXX, XXI, XXXII, XXXV. 
4 



9 



PLAN OF THE LESSONS 21 

6 Lessons : XXXI, XXXIV, VI (Weaving, p. 74). 
4 

10 

7 Lessons: VIII (Weaving, p. 78). 
4 



11 



8 Lessons : IX, XXVIII, XXXIV. 

4 



12 



5 Lessons: XXVI, VII (Weaving, p. 76), 
5 



10 



6 Lesson: VII (Weaving, p. 76). 
5 

11 

7 Lesson : X. 
5 

12 
6 Lessons: XII, XV, XXVIII, XXXIII, XXXIV. 



12 



22 MATHEMATICAL CONSTKUCTION 

2, 1"— XIV, XVII, XXIX. 

2f 2"— IV, VII, XIII, XIV, XVI, XVII, XXVIII, XXIX, 

XXXI, XXXIV. 
2, 3"— III, VIII, XII, XV, XXV, XXVI, XXVIII. 

XXXIII. 
2, 4"— II, IX, XIII, XVI, XXVIII, XXXI, XXXIV. 
2, 5"— XXVI, XXXV. 

2, 6"— XII, XV, XXVIII, XXXIII, XXXIV. 
3 ? i"_xiV, XXXVI. 

3, 2"— VII, XIII, XVI, XXV, XXVI, XXVIII, XXXI, 

XXXIV. 
3 ? 3"_ VIII, XII, XV, XXII, XXV, XXVIII, XXXIII. 
3 ? 4"— IX, XVIII, XXVIII, XXXII, XXXIV. 

4, 1"— XIV, XVII. 

4j 2"_ XIII, XVI, XXVIII, XXXI, XXXIV. 

4, 3"— XII, XV, XVIII, XIX, XXVIII, XXXIII. 

5, 1"— XXIX, XXIV, XXVII. 

5, 2"— XXVI, XXXIV, XXXV. 

6, 1"— XXXVI. 

6, 2"— X, XIX, XXVIII, XXXIV. 



FOLDING 

LESSON I 
Shawl 

Note. — Early lessons should be given without verbal command. Teacher 
should only show what to do. Then let children imitate. 




Material : An oblong piece of paper. 

Give word oblong in this way: 
Show me a long edge of this oblong. 
Show me a short edge of this oblong. 
Show me an edge longer than a. 
Show me an edge shorter than b. 
Show me an edge shorter than c, etc. 
Play the oblong is a shawl. 

Which edge needs more fringe, a or b ? (Teacher points 
to edges.) 

If the fringe on a costs a nickel, what does the fringe on 
b cost? (Ans. More than a nickel.) 

If the fringe on b costs a nickel, 
what does the fringe on a cost? (Ans. 
Less than a nickel.) 

Teacher folds so that the edge a 
exactly touches b. Children imitate. 
Result: Figure 2. 
Children lay theirs on desks. Teacher shows large undi- 
vided surface to them. Play that it is a piece of cloth and 

23 




Fig. X. 



24 



MATHEMATICAL CONSTEUCTION 



repeat questions on relations of surfaces as before. Viz. : If 
it takes an hour to hem this side (pointing to d) will it take 
more or less to hem this (pointing to c) ? Ans. More than 
an hour, etc. 

Fold back oblong and tear off. 

Result: A doll's shawl. 

Surface Seen While Making Shawl 
Draw them on the blackboard to scale of 6" to 1". 




Suggestive comparisons on which to base problems 

A=3 of C. E=2 of C. 

=3 of B. =2 of B. 

=sum of B+E. =D. 

=sum of D+B. =sum of B+C. 

=sum of E+C. 

=sum of D+C. 



FOLDING 



25 



B=i of E. 
=4 of D. 
=i of A. 



D=2 of B. 
=2 of C. 
=E. 
=A-B. 
=sum of B+C. 



Note. — If child does not see the relations of the surfaces, cut pieces 
of paper the exact size of the blackboard figures. Dissect and put the 
pieces together in any necessary form to give the child a clear image of 
the relation of the surfaces. 



LESSON II 



Booklet 



Repeat lesson I but speak of the oblong being a cover for 
the library table, the edges of which are stenciled, hemmed, 
embroidered or fringed. 

When A is obtained the children 
have 2 surfaces each a triangle. Play 
one triangle is a library floor, what is 
opposite surface? (Ans. Another 
floor just as big.) 

If it takes 10 yds. of carpet for the first floor, how many 
yds. for opposite floor? The teacher just points to the sur- 
faces saying: If it takes 10 yds. for this floor, how many 
for this? (Ans. Just as many, 10 yds.) If this costs 
$25.00, what does this one cost? etc. 

Unfold the paper. Besult: A square. 




26 



MATHEMATICAL CONSTRUCTION 



Play this is a rug for the library. 

If the fringe on A costs a dollar, what does the fringe on 
C cost? on Df on B? How many dollars for fringe on A 
and C together? D and B together? 
How many dollars for all sides? 

Fold the square into 2 equal oblongs. 
Here are two oblong surfaces to compare 
as the triangles were compared. 
Result: A booklet. 




LESSON III 



Wall-pocket 
(New relation 1 to 2) 

From an oblong proceed to make a square as in previous 
lessons, but do not ask too many questions on the surfaces 
and lines already talked about. The same relations with 
different lengths of lines and size of surfaces can be found 
in the new folds. 

After the square has been folded into 2 equal oblongs the 
new folds begin. 

Fold so that the short edges touch. 

Result, 4 small squares. Children see only the outside 
ones. 

Teacher points to 1 square and says : If this window 
glass costs a dime, what does this one cost? (Pointing to 
opposite one.) (Ans. One dime.) 

If it costs a quarter? (Ans. One quarter.) 



FOLDING 27 

If putty on one edge costs a penny, what does putty on 
another cost? What does putty on all cost? 

Hold square so that 4 free points point 
upward. Teacher moves one point from A S^K 

to B, thus making a wall-pocket. / N. 

Now child sees a whole triangular sur- <(C X 

face (0) and the opposite side which is a \^ q / 
square. \. / 

Now play C is a piece of glass for a B 

window. Teacher points to opposite surface (the square) 
and says: How many pieces just as big can be cut from 
this? (Ans. 2 pieces of glass.) If this (pointing to C) 
costs a nickel, what does this (pointing to opposite square) 
cost? (Ans. 2 nickels.) 

If it costs a dime? (Ans. 2 dimes.) 

If it costs a quarter? (Ans. 2 quarters), etc. 

Suggestive Comparisons of Surfaces Seen While Making 
Wall-pocket 



Draw surfaces on 


blackboard to any scale large enough to 


be easily seen by pupils. 


Base 


problems on these surfaces. 


A=sum of B+C. 






=E 


=sum of D+E. 
=3 E's. 
=3 Fs 
=3 B's. 






=2 G's 
G=2 H's. 
=4 of F. 
4 of B. 


=6 G's. 






=J of D. 


B=i of A. 






=i of C. 


=4 of c. 






=J of A. 


=4 H's. 






F=2 G's. 



=± of D =4 H's. 



28 



MATHEMATICAL CONSTRUCTION 




LESSON IV 

Soldier's Cap 

Made just like wall-pocket with an additional fold at the 
end; the folding back of the three points which were left as 
the back of the wall-pocket. Eesnlt : 
Soldiers cap. 

Number relations not different 
than Lesson III; but this should be 
made of a differently sized paper, to 
give a new length of lines and size 
of surfaces not relative but actual. 




FOLDING 29 

Play that the large square is a drill ground for soldiers. 

If the ground is 1 block long, how wide is it? (Ans. 1 
block wide.) 

If one company of soldiers can stand next to one another 
on this side (pointing to one edge), how many can stand 
next to one another on this edge (pointing to another edge) ? 
When folded into oblongs these questions: 

If there is room for one company to drill here (showing 
1 oblong), there is room for how many companies to drill 
here (show other oblong) ? After triangle is folded back 
repeat question and get relation 1 to 2. 



LESSON V 
Fireman's Cap 

No new number relation. A review of relations found 
in Lessons III and IV. Use a different size of paper. 

Make soldier's cap; then press the front and back points 
of the cap together. Hold so that 2 free points point up- 
ward. Fold one of them back until it touches bottom corner. 
Eesult : Fireman's cap. 

Play that the oblongs are windows in the engine house. 
Question on time it takes to wash them. 

Play that squares are floors. Question on scrubbing. Or 
play that squares are sheets on fireman's bed. Question on 
hemming of edges; or, in comparing two surfaces, question 
on length of time to iron them. 



30 



MATHEMATICAL CONSTRUCTION 




Play that triangle is a piece of 
rubber for making the hat. Problems 
on cost of it. If this piece of rubber 
(pointing to A) costs a nickel, what 
does this cost (pointing to opposite 
square) ? Ans. 2 nickels. If it costs 
8 cents? Ans. 2, 8 cents. 



LESSON VI 

Envelope 
(New relation 1 to 4 and 1 to 3.) 

Play the square is the top of a desk. 

Compare cost of varnishing to another child's square just 
the same size. 

When folded on a diagonal, play that one triangle thus 
formed is a tile of the school vestibule floor. Compare cost 
of this with opposite triangle. 

Teacher holds her paper 
so that folded edge is down. 
She folds one of the upper 
points (A) to center of folded 
edge. Children imitate. Re- 
sult: Fig. 1. 

B is a whole surface itself. 
Compare it with opposite surface. 




Relation 1 to 4. 



FOLDING 



31 



If B is one tile opposite surface 
is 4 tiles. 

If B costs a dime opposite sur- 
face costs 4 dimes, etc. 

Fold point C to E. Result: 
Fig. 2. 

F is a whole surface, so is G; 
compare F to G. Relation 1 to 2. 




Fi 5 .2 



Fold point D (Fig. 1) to E. Result: Fig. 3. 

Compare F and H. Relation equal. 
Compare F and G. Relation 1 to 2. 
Compare H and G. Relation 1 to 2. 
Compare G with opposite surface. Re- 
lation 1 to 3. 

Of course, give concrete problems to get 
Fig 3 these relations. 




Surfaces Seen While Making Envelope 
Draw them enlarged on blackboard. 




32 



MATHEMATICAL CONSTEUCTION 



Base problems on some of these comparisons 



A=2 of B. 


G =8 of E. 


=2 of G. 


=4 of D. 


=8 of D. 


=i of A. 


=8 of H. 


K=2 H's. 


C=3 of D. 


=F-D. 


=6 of E. 

=6 of J. 

=3 of H. 

=sum of H, D, E, J. 

=sum of K, E, J. 


E=J of D. 

=i of C. 
=i of F. 
=i of K. 


B=sum of C+D. 


D=i of F. 
=H. 


G=B. 


=2 of J. 


=4 of H. 


=1 of C. 


=8 of J. 


=4 of B. 



LESSON VII 



Picture Frame 



Relation 1 to 4; 1 to 5; 1 to 6. 

Fold square into 2 equal oblongs. 

Fold so that short edges of oblongs 
touch. Open. Large square is now 
divided into 4 small squares. (Fig. 1.) 



E 



Ftgt. 



FOLDING 



Fold so that point A touches center 

Fold so that point B also touches center 
(E.) (Result, Fig. 2.) 

Compare surface F with surface G, Fig. 
2. Relation equal. 

Compare surface F with opposite sur- 
face. Relation 1 to 6. 

Compare surface G with opposite sur- 
face. Relation 1 to 6. 

Fold so that point D touches center E, 
Fig. 3. 

Compare H to F or G. Relation equal. 

Compare H to opposite surface. Rela- 
tion 1 to 5. 

Fold point C to center E, Result, Fig. 4 

Compare surface J to opposite surface. 
Relation 1 to 4. 

Fold back point 1 to middle of edge 5. 

Fold back point 2 to middle of edge 6. 

Compare K and M (Fig. 5). 

Compare K and H, etc. 

Fold back point 4 to 8 and 3 to 7. 

Result : Picture frame for a doll's house. 

Base the concrete problems on things 
pertaining to a doll's house. 



E 
D E 



Viq.2. 




P'9 3 





FsgS 



34 



MATHEMATICAL CONSTRUCTION 



The squares and oblongs can be lace curtains for the doll 
house. Question about length or cost of lace for the curtains. 
Some curtains having lace on 2 edges, others on 3 edges. 

Let triangle F (Fig. 4) be a doll's shawl. Let opposite 
surface be a piece of cloth. Ask how many shawls can be cut 
from it. 

Let small square (Fig. 1 before it is opened) be a hand- 
kerchief. Question on cost of lace on edges, or time to hem 
them. 

Surfaces Seen While Making Picture Frame 

Draw them enlarged on blackboard. Suggestive compari- 
sons: 




A=2 of B. 
=8 of C. 
=2 of I. 



A=sum of D+C. 
=sum of H+3 C's. 
=sum of 1+4 C's. 



FOLDING 



35 



A=sum of G+2 C's. 

=sum of D+E+F. 

=sum of G+C+E+F. 
C=l of B. 

=i of I. 

=sum of E+F. 
B=sum of C+E+F. 

=i of A. 

=1. 



D=B+3 C's. 

=7 C's. 
G=6 C's. 

=1+2 C's. 
H=I+C. 

=5 C's. 
1=4 C's. 
F=3 E's. 

=C-E. 



LESSON VIII 




Fig. 1. 



Salt and Pepper Holder 



Fold square on its diagonals. 

Fold points a, h, c, d, to center E as 
in previous lesson, getting small tri- 
angles to compare with opposite surface. 
Eelations 7, 6, 5, 4, respectively. Turn 
Fig. 1 upside down. 



Fold the four points of the new 
square to center E again and turn up- 
side down. 

Insert fingers under points 1, 2, 3, 4, 
Fig. 2, and squeeze into the shape of 
salt and pepper holder. Fig. 3. 



I 2. 
3 V 



Fig. 




36 MATHEMATICAL CONSTEUCTION 

After the holder is made, inquire: 
If one little pocket holds lc worth of 
salt, how many cents' worth Will all 
hold? 
Fig ; . If it holds 2c worth, how many 2c 

worth will all hold? (Ans. 4, 2c 
worth.) 



LESSON IX 

Boat 

See Lesson IV, Soldier's Cap, for beginning of boat. 
Take the two side points of the soldier hat and make them 
touch one another. Now there are 2 free points of a square. 
Fold one of them back to opposite corner. Fold the other 
one back the other way. Result : A little soldier's hat. 
Press the two free points of this together. 
Fold back both in opposite directions to meet opposite 
corner. Eesult : A very small soldier's cap, with 3 points at 
the top. Put fingers on sides of middle point and pull the 
paper. Eesult: Boat. 

When boat is made ask questions about the prow and 
stern of it. Bind them with iron (imaginatively, of course). 
^ ~~~~Z If prow is 5 yds. long, stern is 

\ & 5 yds. long. 

— is If it took 2 days to bind prow 

with iron, it took 2 days to bind 
stern with iron. 

If iron on prow costs $10.00, iron on stern costs $10.00. 




-rr 



FOLDING 



37 



If it takes a week to paint 1 side of ship, it takes a week 
to paint other side of it. 

If it takes $100.00 worth of wood to build one side it 
takes $100.00 worth to build other side. 

Compare cost of painting 1 side of cabin to 1 side of ship 
on the outside. (Eelation of 1 side of cabin to side of ship 
is*). 

Compare cost of painting 1 side of cabin outside and 
inside to 1 side of ship outside and inside, etc. 



LESSON X 



Square Prism 



(To be used for that game in which the teacher says: 
"I am thinking of something which looks like a square prism. 
Guess what it is.") 

Fold a square into two equal oblongs. Open. (Fig. 1.) 
Make edge A touch folded line. Compare oblong thus 
obtained to opposite surface. Eela- 
tion 1 to 3. Make edge B touch folded 
line. Compare oblong thus formed to 
opposite surface. Relation 1 to 2. 
Open whole paper and fold the other 
way into 2 equal oblongs. 

Fold long free edges back to long 
folded edge. Eesult: Large square is 




9 ' 



divided into 16 small squares. 



38 



MATHEMATICAL CONSTRUCTION 



A 


B 


C 


D 
















E 


F 


G 


H 



Cut line 1 and fold back whole upper row of squares. 
(Fig. 2). Compare surface A with 
surface bed. Eelation 1 to 3. 

Cut line 2. 

Compare surface a with surface cd. 

Compare surface cd with opposite 
surface. Eelation 1 to 6. Cut line 3. 

Cut line 4, fold back whole lower 
row. Compare surface ef with sur- s <+ <> 

face gh. ' 5 a 

Compare surface ef with opposite surface. Eelation 1 to 4. 

Compare surface gh with opposite surface. Eelation 1 to 4. 

Cut line 5. Compare surface e with opposite surface. 
Eelation 1 to 8. 

Cut line 6. 

Fold and paste into square prism shape. 

Play game mentioned. 

Concrete problems: In Fig. 1, after a is folded to middle 
fold ask: 

"If this wall (small oblong) needs two rolls of paper to 
paper it, how many rolls will this wall (opposite surface) 
need?" (Ans. 3, 2 rolls.) 

After B is folded to middle fold ask : "If one man paints 
this wall in 4 hrs. ? how long will it take him to paint this 
one (opposite surface) ?" (Ans. 2, 4 hrs.) 

Children should keep square prism in school for future 
comparison with the cube which is to be made next out of 
the same sized paper, as square prism was made. 



FOLDING 



39 



Surfaces Seen While Making Square Prism 

Enlarge and draw them on the blackboard. 

Base concrete problems on some of these comparisons: 



B 



A=2 of B. 
=4 of C. 
= 8 of G. 

=sum of D+C. 

=sum of B+2 Cs. 
B=2 of C. 

=D-C. 

=A-2 Cs. 

=A-B. 
B=sum of C-t-E+F. 

=sum of C+G+H. 

=sum of E+F+G+H. 

=4 of G. 

=§ of A. 
C=i of D. 

=F+E. 



B 



H 



=G+H. 
=A-D. 
=D-B. 

=2 times G. 

=4 times E. 
F=3 times E. 

=snm of G+E. 
0=4 of A. 

=i of B. 
F=C-E. 

=i of D. 
G=2 Es. 

=H. 

=F-E. 

=C-H. 



a]'b[g 


3 
D 


! » ! 

t 1 t 


E 


F 


| 


15 if \(o 

• » ; 
» i i 



40 MATHEMATICAL CONSTRUCTION 

LESSON XI 

Cube 

Fold 16 squares as in previous lesson. Cut off lower row. 
Compare this to surface which is left. If large surface will 
hold a loaf of cake, what will smaller 
surface hold? (Ans. ^ of a loaf.) 

If it takes 3 cups of flour for the 
former what will it take for the latter ? 
(Ans. -J of 3 cups). 

Fold back row of 3 squares. Com- 
pare this to opposite surface in same 
way getting result ^. Then put ques- 
tions the opposite way, getting result 3. 

Fold back other outside row of 3 squares and repeat ques- 
tions, getting answers J and 2. 

Open folds, making large flat surface with 12 small 
squares. Make cut 1. Fold back surface led. 

Compare it with opposite surface. Eelation ^ and 3. 
Make cut 2. Compare cd with opposite surface. 
Through questions get relation J and 4. 
Make cut 3. 

Make cut 4. Fold back ef. Compare with opposite sur- 
face. Eelation ^ and 3. 
Make cut 5. 
Make cut 6. 
Fold and paste into a cube. 

Compare Cube and Square Prism 

How many cubes like this can be cut from the square 
prism ? 



FOLDING 



41 



How many times larger is the square prism than the cube ? 

The cube is what part of the square prism? If the cube 
is a child's building block, how many can be made from the 
square prism? If the little block costs a nickel, what does 
the big one cost? (Ans. 2 nickels). If the prism costs 6c 
what does the cube cost? (Ans. J of 6c), etc. 

Surfaces Seen While Making Cube 

Enlarge and draw them on the blackboard. 

Base concrete problems on some of these comparisons: 





H 



J 



K 



C=i of A. 

=i of B. 



C=J of D. 
=i of I. 



42 



MATHEMATICAL CONSTRUCTION 



C=2 of K. 


D=sum of H+E 


E=J of F. 


I=B. 


=4 of G. 


=2 of C. 


=i of H. 


=4 of A. 


=4 of J. 


=4 of K. 


=i of D. 


=sum of J+K. 


=F-G. 


J=3 of K. 


=D-F. 


=G. 


H=sum of G+E. 


=2 of E. 


=sum of J+E. 


=C+K. 



LESSON XII 



Cradle 



Cut off lower row of squares. Compare this with large 
surface. If small surface is enough 
cloth for a pillow case for baby's bed, 
what is large surface? (Ans. Enough 
for 3 pillow cases). Make cut 1. 
Fold back b-c-d. Compare it with 
large surface. Eelation 1 to 3. Fold- 
ing back a-e-f and comparing it to 
what is left on opposite side gives the 
relation 1 to 2. Make cuts 2, 3 and 4. 
Fold and paste into box shape. Eock- 
ers to be made free-hand from row of 
squares cut off originally, then pasted 
on. 



1/ 1 1 

A B ; » D 

E ' • ' 
i ' ' 


2 1 i 

F j . 4 

1 




: ; 

1 » 
1 1 • 



^ 



FOLDING 



43 



New Surfaces Seen While Making Cradle 
(For other surfaces see lesson on Cube). 
Enlarge and draw them on the blackboard. Base concrete 
problems on the comparisons: 







A 










B 






A=4 of D. 


B = 3 of A. 


B=2 of A. 


=i of B. 


=3 of C. 


=2 of C. 


=C. 


C=i of D. 


=sum of A + C 


B=sum of D+C. 


=i of B. 


=B-C. 


=sum of D+A. 


=B-D. 
LESSON VIII 


=B-A. 



Chair 

Fold 16 squares. Cut off one row. Make cut 1. Fold 
back ah. Compare this with opposite surface. Play it is a 
blackboard. Pointing to opposite 
surface, this blackboard is how many 
times as large? (5 times.) If it 
takes one boy 1 minute to clean it, 
how long will it take him to clean the 
other? (5 minutes.) How many 
boys could clean it in 1 minute? (5 boys.) Make cut 2. 
Fold back c-d and repeat comparisons — getting result 4 in- 



! „ 1* 

A • B | H E 


— I 

C I D 


1 

> 1 

1 

g] f 



44 



MATHEMATICAL CONSTEUCTION 



stead of 5. Make cuts 3 and 4. Fold back e and /. Com- 
pare c-d to what is left on opposite side. Result : Relation 3. 
By reversing questions get relations 1/5, 1/4, 1/3, respectively. 

Paste e under h. 

Paste / under g. 

Paste a under g and /. 

Paste c-d on i-j to strengthen back. Then chair is made. 
Legs may be cut out free-hand. 



B 



o 



D 



Chair 

Enlarge and draw on blackboard 
new surfaces not seen in previous les- 
sons. 

Give concrete problems based on 
these comparisons : 

A=l/5 of B. 

=1 of C. 

=i of D. 
B=sum of C+A. 
=sum of D+2 A's. 
C=4 of A. 

=B-A. 

=D+A. 
D=3 of A. 

=C-A. 
=B-2 A's. 



FOLDING 



45 



LESSON XIV 
Buggy 

From two equal oblong papers fold to get squares. From 
the little extra oblongs cut wheels and shafts free-hand. 

Fold one square into 16 small 
squares. Cut off 1 row. 

Compare this to 12 squares 
which are left. 

Base problems on things seen 
when driving. 

If a requires 10c worth of grass 
seed, b requires 3, 10c worth. 

If it takes 2 days to put a cement walk on a, it takes 3, 
2 days to put one on b. 

If a yields 5 bu. corn b yields three 5 bu. 

If b yields 3 pk. potatoes a yields J of 3 pk., etc. 

Make cuts 1 and 2. Fold back left row and compare to 
opposite surface. Relation 1 to 3. 

Make cuts 3 and 4. Fold back right row and compare 
to left row (equal). Also compare to opposite surface. (Rela- 
tion 1 to 2.) Fold and paste into box shape. Do same with 
other square. 



it 


a • ! 





46 



MATHEMATICAL CONSTRUCTION 



Compare volumes of two box shapes. If one holds -J pk. 
oats for horse the other holds J pk., etc. 

Cut one of these as indicated in Fig. 2. 

Insert and paste this into other box shape. Use toothpicks 
for axles, adjust wheels and paste on the shafts. 



BASKETS 

This is a series of basket patterns more than anything 
else, just to show what a variety can be made from 16 squares 
cutting either on a straight fold or a diagonal. I have par- 
tially shown the variety of surfaces to be compared, but have 
not inserted any concrete problems, though of course they 
should be given whenever any comparing is done. 



• 


,14... 


i 
— i- - 

i 

i 

i 

T 


i * 
_• ' 

i ! 



• 
i 

if. 

i : 

i 


5 i 

■r-i 
... 




i i 
i i 

. !__•_ -J 



LESSON XV 

Ordinary box shape, based on 16 
squares, straight cuts. 



LESSON XVI 

Oblong box shape. 1 row of squares 
cut off, straight cuts. Cutting for flaps 
different to give differently shaped sur- 
faces on which to base problems. 



FOLDING 



47 



i ; 
t , ' 

i . 

i 


; i ; i i 
I •_ . • I 


■ * i ' ■ 
' ! • i 



LESSON XVII 



16 squares; 2 rows cut off; straight 



cuts. 




Fig. 3 is the result 
when the free corners are 
cut off. 



LESSON XVIII. 



5 

, I 

I 

I I 

I I 

: ; 

•j-.l. .;...;- 

, i , . 




Like a cube 
with two sides 
cut out. 



48 



MATHEMATICAL CONSTRUCTION 



LESSON XIX 



Box shape; 9 squares; straight cuts. When bottom row 
is cut off number relation is 1 to 3. 

When right row is cut off number 
relation is 1 to 3. 

Make cuts a b. Fold back 2, 3, and 1. 
The sum of 2 and 3 can be cut how many 
times out of opposite surface? Sum of 
2 and 3 is what part of opposite surface? 
Make cut C and fold back 4. Compare to 
opposite surface. Relation 1 to 5. Make cut d. 
Fold and paste into box shape. 




LESSON XX 





• 


\ 
\ 


• 
• 


* 

£ - - 

X 

\ 
\ 






i 


• 


1 


k 


y 


1 

__ J 




Make cuts 
indicated and 
fold on the 
d iagonals 
seen. Paste 1 
on 2. Make 
handle of 3 
strips cut 
free - hand. 
Eesult : 
Fig. 2. 



FOLDING 



49 



LESSON XXI 




Eight squares. Di- 
agonal cuts. 

Cut off free cor- 
ners if you so desire. 



LESSON XXII 




Six squares. Diagonal cuts. After 
a and b are cut out, the sum of a and 
&=J of whole surface left. 

a is £ of whole surface. 

b is i of whole surface. 



iff 



Open. 

Eold back on line e-f. 



Whole surface=4 of a or 

4 of & or 

2 times the sum of a and b. 
Fold on horizontal dotted line. 
Now a is J of surface seen. 
Now b is | of surface seen. 
Sum of a and b equals surface seen. 
Cut a on horizontal dotted line. 
Triangle 1 is J of b. 
B=2 of triangle 1. 



50 



MATHEMATICAL CONSTRUCTION 



Compare b to surface seen (^). 
Fold on horizontal dotted line again. 
Compare triangle 1 to surface seen (^). 
Open. 

Fold on lines g-h and e-f. 
Compare b to surface seen (^). 
Compare sum of A 1 and A 2 to surface seen (\). 
Compare sum of A 1 and A 2 and b to surface seen 
equal. 

Fold and paste. 



LESSON XXIII 

Fold back whole upper row after cuts 1, 2 are made. 
Compare Fig. a-b-c with opposite surface. Relation 1 to 3. 

Make cut 3. Fold back whole left row. 

Compare surface a-b-c to surface d-e-f. Compare d-e-f 
to opposite surface. Relation 1 to 2. Make cut 4. Compare 
g-h-i to opposite surface. They are equal. 

Make long edge of triangle / touch right edge of triangle 
g and paste. Treat other corners similarly. 

Cut two long strips free-hand for handles. 



E ; ; 


i 1 




.-i ( 




1 


i • • 



FOLDING 



51 



LESSOR XXIV 



/ 




2 










J I [ ' M 


* 

1 
1 


3 


« ! ! 

L— ! 1 J — . 



Make cuts 1 and 2. 

Fold back surface a-b-c-d. 

Compare this with opposite sur- 
face. 

Relation 1 to 3. 

Make cuts 3, 4. 

Fold back surface e-f-g-h. 

Compare this surface to surface 
a-b-c-d (equal). 

Compare it to opposite surface (relation 1 to 2). 

Fold back i-j-k. 

Compare i-j-k to opposite surface (relation 1 to 2). 

Fold back l-m-n. 

Compare it to i-j-k. 

Compare it to opposite surface (equal). 

Make long edge of A a touch lower edge of A i. Paste. 
Treat other corners similarly. Cut off points which pro- 
trude. Cut handle free-hand. 

This makes a nice basket for Eed Riding Hood, or a 
shopping basket. 



<c 



i 



52 



MATHEMATICAL CONSTRUCTION 

LESSON XXV 



f 


Make cuts 1, 2. 
a Fold back whole upper row. 


^I_b_;_c^ 


Compare a-b-c-d to whole opposite 


K 1 • 1 


surface (%). 


--I-;--*— 


Cut 3, 4. 


l! : ! 


Fold back e-f-g-h. 


%If : « ?? 


Compare surface a-b-c-d to surface 


J e-f-g-h (equal). 






Compare i to surface a-b-c-d. 



Compare i to /. 

Compare sum of i and / to surface a-b-c-d. ($). 

Compare e-f-g-h to opposite surface. (Eemember that 
top row is turned back). Eelation 1 to 3. 

Fold back i-k-l-m; also o. 

Compare i-Tc-l-m to a-b-c-d; or e-f-g-h (equal). 

Compare i-k-l-m to opposite surface (J). 

Fold and paste so that longest edge of triangle a touches 
lower edge of triangle i. Treat other corners in like manner. 
Cut off protruding points. Eesult: workbasket for sewing 
materials. 



FOLDING AND WEAVING 



LESSON I 



Material: Paper 6"x9". 

Fold and tear so that a 6-inch square is obtained. Color 
remaining oblong, which is 6"x3", on one side with crayola. 
Compare the two surfaces, giving problems on size and 
cost of rugs; also fringe on edge. Fold 6 inch square into 
16 small squares. Then fold into two equal oblongs. 
Make cuts indicat- 
ed at a and b on fold- 
ed edge. Fig. 1. Open. 
Weave with the col- 
ored strip. Eesuit : 
Fig. 2. 





LESSON II 






4 B 



Repeat Lesson 7, but tear off 
one row of the sixteen squares 
making cuts a-b (Fig. 1) on 
folded edge. 

Weave with colored strip. 

Result: Fig. 2. 




53 



54 



MATHEMATICAL CONSTEUCTION 



LESSON III 



Paper same as Lesson I. 

Fold and tear to get 6-inch square. Color with crayola 
the oblong which is left. Then divide it into two equal 
oblongs by folding and cutting or free cutting. 

Fold 6-inch square into 16 small squares. Open. 
Fold into 2 equal oblongs. 

Make cuts indicat- 
ed at a, ~b, c on folded 
edge (Fig. 1). Open 
and weave. Eesult : 
2. 



Fig. 




LESSON IV 



Paper 6"x9". 

Make 6-inch square and color oblong which is left. Cut 
this oblong into 4 equal oblongs by folding and cutting on 
fold or free cutting. Fold square into 16 small squares. 
Open. Fold into 2 equal oblongs. 

Make cuts indicat- 
ed at a, h, c on folded 
edge (Fig. 1). Open 
and weave. Result : 



Fig. 2. 





• 


— r 
i 




-r— 

i 
i 




" " "in 


■ 




.j... 


- 


-- 


"■ 


: 


■ 


■ 


•- 


- 


4_.j 

j ! 


■ 


i 
• 


-- 



FOLDING AND WEAVING 



55 




LESSON V 

Paper 6"x9". 

Fold so that a 6-ineh square is obtained. Color the oblong 
which is left. Divide it into 2 equal oblongs. Fold square 
into 16 small squares. Open and fold into two equal oblongs. 
Make cuts indicated at a, h, c on 
folded edge (Fig. 1). Now with free- 
hand cutting (judging with the eye, there 
is no fold to cut on) 
make cuts d, e. Open 
and weave. Eesult : 
Fig. 2. 



LESSON VI 

Paper 6"x9". 

Fold so that a 6-inch square is obtained. Color oblong 
which is left. Cut it into 4 equal oblongs. Fold square into 
16 small squares. Open and fold into 2 equal oblongs. Make 
cuts indicated at a, b, c, on folded edge. 

Eesult: Fig. 1. 





Make cuts d and e free-hand. Weave. 
Result: Fig. 2. 



A V D £ C 



Color the border if you care to. 




56 



MATHEMATICAL CONSTRUCTION 

LESSON VII 



Two 8-inch squares contrasting colors. 
Fold the one which is to be used for strips into 8 equal 
oblongs. Cut on folds. Fold the other square into 8 oblongs 
one way. Open and fold into 8 oblongs the other way. 
Open and fold into 
2 large equal oblongs. 
Make cuts indicated 
in Fig. 1 on folded 
edge. Open and weave. 
Result: Fig. 2. 




LESSON VIII 



Two 8-inch squares contrasting colors. 
Fold the one to be used for strips into 4 equal oblongs. 
Cut. Then divide one of those oblongs 
into 2 equal smaller oblongs by free 
cutting or folding. 

Fold the other 8-inch square into 
4 equal oblongs. 

Fold the two free edges back to the 
opposite folded edge. Open. 

Fold the other way in same manner. Make cuts and 
weave with a narrow strip first, two wide ones next and a 
narrow one last. Eesult : Fig. 1. 




FOLDING AND WEAVING 



57 



LESSON IX 



Two 8-inch squares contrasting colors. 

Fold the one to be used for strips into 4 equal oblongs. 
Divide 2 of these lengthwise into 2 equal oblongs by free 
cutting or folding. 

Fold the other 8-inch square into 4 
equal oblongs. Fold the 2 free edges 
back to meet opposite folded edge. Open 
and fold the other way in same manner. 
Make cuts and weave with 2 narrow 
strips first, then a wide one, then two 
more narrow strips. Eesult: Fig. 1. 




LESSON X 



Two squares, contrasting colors, any size that is large 
enough. 

Fold the one to be used for strips into 2 equal oblongs. 
Divide one of these into 2 smaller oblongs. Divide one of 
these 2 small ones into two equal smaller oblongs. Cut. 

Fold the other square into 4 equal oblongs. 

Fold the 2 free edges back to oppo- 
site folded edge. Open and fold the 
other way in same manner. Make cuts, 
open and weave, using two narrow strips 
first, then the wide one, then the other 
two narrow ones ; going over one and un- 
der two. Eesult : Fig. 1. 




58 



MATHEMATICAL CONSTEUCTION 



LESSON XI 






Two squares, contrasting colors. 

Cut the square to be used as strips into 4 equal oblongs. 
Bisect two of these lengthwise and then divide 2 of the nar- 
row oblongs thus obtained into 2 equal smaller ones. 

Fold the other square into 8 equal 
oblongs. Open and fold the other way 
into 4 equal oblongs. Fold back the 2 
free edges to the opposite folded edge. 
Open so that the paper is folded into 
halves. 



Make cuts indicated from folded edge 
(Fig. 1). Open and weave, going over 
two and under one, using one middle 
sized strip first; then 2 narrow ones, then 
1 wide one, then 2 narrow ones, then 1 
middle sized one. Result: Fig. 2. 




LESSON XII 



Two squares, contrasting colors. 

Cut the square to be used as strips into 4 equal oblongs 
Divide each one of these lengthwise into 4 equal oblongs 
Sixteen narrow strips are the result. 

Fold the other square into 8 equal 
oblongs. Open and fold the other way 
into 4 equal oblongs. Fold back the two 
free edges to meet opposite folded edge. 
Open so that paper is folded into halves. 



FOLDING AND WEAVING 



59 



Make cuts indicat- 
ed from folded edge 
(Fig. 1). Open and 
weave over 3 and 
under 1. 

Fig. 2 shows two . . 
results of weaving this 
way. 




FREE CUTTING AND WEAVING 



The child has dealt with the relations 1, 2, 3, 4, equal, 
\, l, % in the folding lessons; now he is ready to do some 
free cutting. He should try to cut a piece of paper into 2, 
3 or 4 equal pieces, as the case may be. 



LESSON I 

Material: Paper 6"x9". 
Fold and cut to get 6-inch square. 
Color oblong, which is left with crayola. 
Cut into 2 equal oblongs. 
Fold square into 2 equal oblongs. 

Hold it so that folded edge is down. Now cut so that 
folded edge is divided into 2 equal parts and cut is made half 
way to top of paper. (Fig. la.) 

Divide left half of folded edge into 2 
equal parts the same way (&). Divide 
right half same way (c). Open and 
weave. Eesult same as Folding Lesson 
III. 

60 




FREE CUTTING AND WEAVING 
LESSON II 



61 




Eepeat I, but cut the oblong which 
is for strips into 4 equal parts instead 
of 2. 

After folded edge of mat is divided 
into 2 equal parts, divide each half into 
4 equal parts, cutting half way to upper 
edge. Weave. 



LESSON III 



Material: Two 5-inch squares, contrasting colors. 

Fold the one which is to be the mat into 2 equal oblongs. 

It is time now to show the children how to cut a strip for the 

border, along each end about J an inch 

wide and within half an inch of the top 

(a, b). 

Now divide the folded edge between 
a and b into 3 equal parts and make cuts 
A q d c > d, stopping half an inch from the top. 

Open. 

Cut an inch strip from the other 5- 
inch square. Teacher cuts first. Child 
imitates. This 1-inch strip is discarded 
after some concrete problems have been 
given causing child to compare it to sur- 
face which is left. (Relation 1 to 4.) 




62 



MATHEMATICAL CONSTRUCTION 



Cut this large surface into 3 equal oblongs, making cuts 
parallel with long edges of surface. Weave. 



LESSON IV 




into 3 equal parts. 

ones, then other wide one 



Material: Two 8-inch squares, con- 
trasting colors. 

Make foundation same as in Lesson 
III. 

When 3 strips have been cut, divide 
one of them into 3 equal oblongs cutting 
lengthwise. Divide middle strip of mat 
Weave, using wide strip first, then 3 small 



LESSON V 



Material: Two 9-inch squares, con- 
trasting colors. 

Foundation same as Lesson III. 

Divide two of the three strips for 
weaving into 3 equal oblongs each. 

Divide the left and right strips of the 
mat into 3 equal parts, leaving middle 
one as it is. 
Weave, using 3 narrow strips first, then the wide one, 
then 3 more narrow ones. 




FKEE CUTTING AND WEAVING 



63 



LESSOR VI 



Material : Two 10-inch squares, con- 
trasting colors. 

Fold one square into 2 equal oblongs, 

cut the half-inch strips for border (a, b) 

from folded edge. Divide folded edge 

into 2 equal parts at c. 

Divide a-c into 3 equal parts. Divide c-b into 3 equal 

parts. 

Cut an inch strip from other square. 
Divide large surface into 2 equal ob- 
longs, cutting parallel to long edges. 

Cut each of these into 3 equal parts. 
Weave in any desired way. Over one, 
under one ; over two, under one ; over two, 
under two; over three, under one; over 
three, under two. 




LESSON VII 



Material: Two oblongs 8"x5", 
contrasting colors. 

Fold the one which is for the mat 
so that the long edges meet. Cut as in- 
dicated at a, b, making strips for border. (Fig. 1.) 

Divide the fold between a and b into 5 equal parts, cut 
ting to half an inch from top. 



B 



64 



MATHEMATICAL CONSTEUCTION 



Cut an inch strip from one of the long edges of the other 
oblong from which strips are to be cut. 

Now divide this into 
5 equal oblongs, cutting 
parallel to long edges. 

Cut 2 of these ob- 
longs into two smaller 
equal oblongs each. 
Weave thus: one wide 
strip, two narrow; one 
wide, two narrow, one 
wide. Eesult: Fig. 2. 




f 



FEEE CUTTING AND WEAVING 



65 



WALL PAPER MADE BY FREE-CUTTING 



Fig. 1. Strips any width, 
pasted on background at a 
distance from one another 
equal to their own width. 
Border, little oblong strips, 
same width as wall paper 
strips, laid as shown. 

Fig. 2. Strips any width, 
pasted on background so that 
distance between strips is 
twice the width of the strips. 
For border, cut strips same 
width as other strips, but 
have relation of length of 
small oblongs to be cut from 
these strips 1 to 2. Paste 
border as suggested. 




1- I 



1— I — I — I — I- 



T t i - a 



>>o:<X4KOXK< 



Fig. 3. 



66 



MATHEMATICAL CONSTRUCTION 



Fig. 3. Kelation of strips to background space 1 to 3. 
Border made from equal squares, which have one corner cut 
out. Eelation of this corner to part left is 1 to 3. 




inn 



Fig. 4. 



Fig. 4. Strips grouped. Kelation of narrow to wide 
strips 1 to 4. Eelation of spaces seen in background 1 to 4. 

Width of oblongs in border twice that of narrow strips. 
Oblongs placed so that one-fourth of one touches J of next one. 

Fig. 5. Eelation of strips to background 1 to 4. Figures 
on background and in border are equal squares cut on a 
diagonal. Strips for border are half the width of the other 
strips. 



FREE CUTTING AND WEAVING 



67 




Fig. 5. 



PAPER RINGS MADE BY FREE CUTTING 



Make curtains or Christmas tree ornaments of paper rings 
cut in a definite way. Take an 8-inch square for instance. 
Cut it into 2 equal parts free-hand. Divide each one of these 
into 4 equal parts. Now you have 8 1-inch strips 8 inches 
long to be divided in one of several ways according to the size 
of rings desired. 

One way — Bisect them vertically and horizontally. 

Another way — Trisect them vertically and horizontally. 

Another way — Quadrisect them vertically and horizon- 
tally. 

Another way — Trisect them vertically and quadrisect 
horizontally. 

Another way — Bisect them vertically and trisect horizon- 
tally. 



MEASURING AND WEAVING 

LESSON I 

Bank Decoeation 

(For bank in Buler Lesson IX, page 93) 

Material : A 4-inch square and an oblong 4"x2" of 
contrasting color. 

Rulers: 1"2"3"4". 

Compare oblong to square, giving problems. (Relation 
1 to 2.) 

Compare rulers and estimate length and width of papers 
with them. 

Divide the 4-inch square into 16 one-inch squares, using 
rulers 3"2"1", as shown in regular measuring lessons. 

Fold it into 2 equal oblongs. Make 
cuts indicated on folded edges. Cuts, 1, 
I J 3, 5 are on the lines. Cuts 2, 4 are half- 
way between. 
i 2 > V >~ Cut ti' ie oblong 4"x2" into 4 equal 

oblongs lengthwise. Weave. 
Make six of these and paste them on the outside surfaces 
of the bank made in Ruler Lesson IX, page 93. 

68 




MEASURING AND WEAVING 



69 



LESSON II 



Calendar Back 



Material : One sheet colored paper 
8"x5" ; one sheet paper 7"x5", of contrast- 
ing color, from which to cut strips. 

Rulers: 1"2"3"4"7". 

Tell children length and width of paper. 
Let them find out with their rulers what 
sums equal the width, 5"; or the length, 8" 
or 7". They will see. 



3 2 

2 2 

- 1 

5 — 



-H 



8 — — 



5' 1"=5" 

4' 2"=8" 

Now on long edges of paper 8"x5" with 7-inch ruler 
make marks and draw line a-b. 

Make marks c, d with 1-inch ruler and draw line. 

Fold lower edge to line a-b. Open. 

Fold upper edge to line c-d. Open. 

Use 4-inch ruler to make marks g, h, measuring from left 
on upper and lower edges. Draw line g-h. Now with 3-inch 
and 2-inch rulers respectively make the two inner long-lines. 

With 1-inch ruler make marks e, f, and draw line. 

Fold lower to upper edge, having lines on the outside. 

Cut from folded edge up to the fold near the top on every 
line which was drawn. Now divide each one of these free-hand 



70 



MATHEMATICAL CONSTRUCTION 



into two equal parts, stopping at the fold near the top. 
mat is now divided into 10 half-inch strips and ready 
woven. 

To make the strips divide the 
7"x5" paper into 7 one-inch strips, 
using single unit rulers to make 
points on the long edges, and draw 
lines connecting them. 

Divide each of these by free-hand 
cutting into 2 equal parts length- 
wise. Weave. Fig. 2. 

On this paste a little art picture 
and a calendar. Braid 3 8-inch 
strands of raffia together, slip it 
through two holes punched at the 
top, tie a knot, and calendar is ready 
to be hung. 



The 

to be 




LESSON III 



Blotter 



Material for mat, 2 sheets of paper 7"x3", contrasting 
colors. 

Rulers: 6"5"4"3"2"1". 

Make the vertical dotted lines in Fig. 
1 with these rulers, using the 6-inch ruler 
first to make the line nearest the right; 
then the 5-inch ruler to make the next 
line to the left of that one, etc. 



f, s .i 



MEASURING AND WEAVING 



71 



Using the long 6-inch ruler first and then laying the 
5-inch ruler in exactly the same way, let the child see the 
exact difference between the two. If the short ruler were 
used first and then the longer one there would be an impres- 
sion that the shorter one was covered up, and relationship 
between the two could not be seen. In building, we do not 
pile large blocks on top of small ones or the structure would 
be unstable and fall. 

Take these little rulers from 1-inch to 6-inches; pile them 
one on the other in regular order from 1 to 6, with the 1-inch 
underneath. What do you see? Well, nothing worth while. 
Now pile them the other way : 6-inches at the bottom and 
1-inch on top. See the regular steps from 1 to 6. There is just 
as much difference between 6 and 5 as there is between 5 and 
4 ; 4 and 3 ; 3 and 2 ; 2 and 1. The child feels these relation- 
ships if the rulers are handled in this way — always making 
the large measurement first and the small one on top of it. 

Fold so that the lower touches the upper edge 
and lines can be seen. 

Now begin to cut on each line on the folded edge. 
Now divide each of these parts into 4 equal parts and 
finish all cuts, cutting until the distance from the top 
is exactly the width of the little parts. 

Now, on the other paper draw the two vertical 

lines in Fig. 2, using the 2-inch and 1-inch ruler to 

make marks and 6-inch ruler 

to draw lines. Cut on lines. 

Subdivide each of these 

strips, cutting free-hand, into 

4 equal long strips. 

Weave. 
Ttf - 3 



f-Z 



Trrn-i[|!:::,:i|-ij.;ht.!.:.LL.: 


Pa-a-n-Dl 


■:;^B"DI 1 B.;B. ,: 



72 



MATHEMATICAL CONSTRUCTION 



3>3>S 



Formula for weaving Fig. 3. 

Over 3, under 3 1 Repeat 

Over 3, under 3 13J 

Over 1, under 1 I times. 
Formula for weaving Fig. 4. 

Over 4, under 4 "1 Repeat 

Over 4, under 4 L3J 

Over 2, under 2 | times. 
On top of this mat place a piece of transparent celluloid, 
underneath two or three sheets of blotting paper. Punch a 
hole in each corner and insert brass-headed fasteners. This 
makes a pretty blotter. 




LESSON IV 
Napkin Ring 



Two pieces of paper, contrasting color, 7"x2". 
Rulers: 1"2"3"4"5"6". 

Divide one piece of paper into 2 equal oblongs lengthwise, 
using the 1-inch ruler to make marks and the 6-inch ruler to 
draw line. Fold so that short edges touch. Cut a little dis- 
tance from folded edge on line. Now divide each half of this 
folded edge into 3 equal parts by making cuts. Continue all 
these cuts until the distance from the top equals the width 
of one of the strips. Open. On other 
paper draw lines indicated in Fig. 1, using 
6"5"4"3"2"1" mlers respectively to do it. 
Cut on these lines. 



ti*-i. 



MEASUKING AND WEAVING 



73 



Divide each one of these strips into 3 smaller equal ones, 
cutting lengthwise free-hand. 
Weave in any desired way. 




1 



TT,<y_ 2. 



. 



Figs. 2 and 3 show two different 
patterns. 

Now put transparent celluloid 
on top and opaque celluloid or stiff 
pretty paper beneath. Punch 2 
Sew through 



holes at each end. 

these holes to keep 

the pieces together. 

Now roll into ring 

form and slip a 
pretty ribbon or cord through the holes and tie in a bow or 
knot. 



7T, *- 3 




LESSON V 
Telephone Pad 

Material: One sheet colored paper 
9"xl2", one sheet white paper 9"xl0". 

Rulers: 1"4"5"8". 

On paper 9"xl2" draw lines indi- 
cated, using rulers 8"5"4"1" respect- 
ively. Fold so that lower edge touches 
upper one and lines can be seen. Cut 
on lines from folded edge up to the one 
which indicates an inch from top. 

Divide middle one inch strip into 3 
equal parts, cutting free-hand up to f" 
cross line 1 inch from top. Open and press flat. 



if r" 8- 



74 



MATHEMATICAL CONSTRUCTION 






On 9"xl0" paper use the 
1-inch ruler on left and right 
long edges to mark them of! 
into inches. Draw lines with 
a long ruler. Cut on the lines. 
Now we have 10 strips 9"xl". 
Divide 5 of them into 3 equal 
parts lengthwise and weave as 
pattern shows. 

If this mat is not stiff 
enough paste cardboard under- 
neath. Punch 2 holes in top, 
and hang with a piece of braid- 
ed raffia or cord. 



LESSON VI 
A Wall-pocket for Letters 



Two pieces stiff paper, contrasting colors, 5"xl0". 
Rulers: 1"2"3"7"10". 

Combinations not seen by the children before in this book 
are shown in this lesson : 



3 3 

7 2 
— 2 
10 — 

7 



3 

2 
3 

2 

10 



MEASURING AND WEAVING 



75 



AG EC 



BHFD 
F'3 < 



Measuring with 1-inch ruler from upper 
and lower left and right corners and drawing 
line with 10-inch ruler, draw a-b, c-d. (Fig. 1.) 

Measure e-f with 3-inch ruler. 

Measure g-h with 2-inch ruler. 

Fold so that lower touches upper edge and 
lines can be seen. Cut on each line from folded 
edge, until half an inch from top is reached. 
Divide each of these parts except the middle 
one into two equal parts. 



Collect these rulers and use 
a different set for the other 


2" 

ft 

6 

3" 




paper. Give children 2"4"6" 
8"10" rulers. 




They see : 

2 2"= 4" 6 2 2 2 




3 2"= 6" 4 8 4 2 

4 2"= 8" 

5 2"=10" 10 10 6 4 






2 4"= 8" 







Fi^^L 



Draw lines indicated in Fig. 2, using rulers 2", 4", 6", 
8". Cut on these lines. Divide each one of the oblongs into 
4 equal pieces, cutting lengthwise, free-hand. Weave. 

With 7-inch ruler measure down on long edges and fold 
on line. See Fig. 3. 



76 



MATHEMATICAL CONSTKUCTION 



Place and paste ends of strips under border after mat 
has been folded on this 7-ineh line. Punch holes and lace as 
indicated. 




Ff^.3 



LESSON VII 

Circular Woven Basket 

Material: An oblong ll"x2"; an oblong ll"x4", con- 
trasting colors; a 3J-inch square for base. 

Rulers: 1"2"5"6"9"10". 

New combinations of numbers are seen when paper 
ll"x2" is given to children and they are told to estimate 
the length of it. Not having the 11-inch ruler in their 
hands there is no harm in saying: "This paper is 11 inches 
long. Find out what rulers put together equal 11 inches, the 
length of the paper." They will see: 
10 5 9 
16 2 



11 11 11 



MEASUEIXG AXD WEAVING 



77 



In addition to these combinations in ruler comparisons, 
they get : 

5 9 5 
1 1 5 



6 10 10 

Bisect this oblong (ll"x2") lengthwise, using 1-inch 
ruler to make marks and 10-inch ruler to draw line. Fold 
so that short edges touch and line is visible. Cut on line 
from folded edge for a short distance. Now divide each part 
into 3 equal parts. Cut all lines up until a distance from 
the top is reached equal to the width of one of them. 

On paper ll"x4" make marks 
and draw lines with rulers indi- 
cated in Fig. 1. Fold so that 
upper edge touches dotted line. 
Cut on fold. Save this piece for 
handle. Now cut on all the ver- 
tical lines. Divide every strip into 3 equal parts, cutting 
lengthwise, free-handed. Weave, letting twice as much of the 
strips protrude on one side as on the other. After the mat is 
woven and pasted into circular shape, these 
smaller protrusions should be turned in. 
They are the flaps, which are pasted on to 
the 3£" square. Trim off protruding parts 
of square. The longer protrusions are 
turned outward and cut into any desired 
decorative form. Q*«"»"t*i> 




78 



MATHEMATICAL CONSTRUCTION 



LESSON VIII 



Needle-book 



Eulers: l"3"4"7" Q 



Material: Two sheets stiff paper ll"x4", contrasting 
colors; 1 sheet thin lining paper ll"x4"; one or 2 oblongs of 
cloth 10"x3"; 10 inches of ribbon for fastening at hinge. 

10". 

On background paper draw 
D lines a-b, c-d (Fig. 1), measur- 
ing with 1-inch ruler from 
B upper and lower edges, or with 
3-inch and 1-inch ruler re- 
Fold so that short edges meet 
and lines can be seen. Cut from folded edge on the lines 
until the distance from the opposite end is \ the width of the 
narrow strips. Divide each of these strips into 3 equal parts, 
cutting parallel to other cuts. Give the other paper to the 
children. Tell children length of the paper (11")- Let them 
find out with their rulers what sums equal 11 inches. They 
will see : 
7 8 
4 3 4 " 



spectively from upper edge. 



?'V 



11 11 

In comparing rulers they will see : 

3 3 7 3 

4 7 11 



T,; r -a 



10 



MEASURING AND WEAVING. 



79 



Draw lines indicated in Fig. 2, measuring with rulers 
10"8"7"4"3"1", respectively, from left edge. Cut on the 
lines. Divide each of these pieces into 3 equal parts, except 
the large middle one. Divide this into 2 equal parts. Weave. 
Paste down all the edges of the strips, and paste lining paper 
on back. Fold so that short edges meet. Insert one or two 
pieces flannel (edges pinked). Punch 2 holes on fold and 
lace with ribbon. 




T ' '> 3 



MEASURING 

Introduction of 

SINGLE UNIT RULERS 

LESSON I 

Scissors Holder (8-inch ruler) 

Materia] : Use grey school paper, which is 9"xl2 
Teacher goes to wall with her paper, 
lays her 8-inch ruler on it as indicated in 
Fig. 1, then draws a line the whole width 
of the ruler at point marked a. Children 
imitate at their seats. (Children must 
keep their papers in the same position on 
the desk until all ruler work is done.) 

Second step: Teacher lays 8-inch ruler on lower edge of 
the paper and draws a 1-inch line at b. 
Children imitate. 

Teacher draws line with her ruler connecting a and b. 
Children imitate. 

Teacher cuts on line. Children imi- 
tate. Now let children compare the two 
oblongs X and Y. 

How many sheets of cardboard as large 
as y can be cut from xf (Ans. Two 
f'* a sheets.) If this (showing y) costs 5c, 

80 



MEASURING: SINGLE UNIT EULEES 



81 



what does this cost (showing x) ? (Ans. Two 5c.) 



(Ans. 



If x 

Half 



costs a dime, what part of a dime does y cost? 
a dime.) Now teacher lays 8-inch ruler 
along left edge of large oblong from upper 
left corner as indicated in Fig. 2. Draw a 
line the width of the ruler at c. 

Children imitate. 

Now lay ruler on right edge from upper 
right corner in same way, drawing line at d. 
Children imitate. 

Draw line from c to d. Children imitate. 

Cut on this line. Eesult : 8-inch square. 

This is enough ruler work for a first 
lesson. Now fold into two equal oblongs. 
Open. Make point a touch middle of top 
line at b. (Fig. 3.) 

Now make c touch d. Eesult: Fig. 4. Cut on dotted 
line. Scissors holder now needs only to be pasted. 




LESSON II 
Mayflower (8-inch and 4-inch rulers) 

Make an 8-inch square as in previous lesson, but use white 
paper. At the end of the lesson let the children use crayola 
to color the hull of the boat black. The white sails will make 
a pretty contrast. 

Call the 8-inch square the tablecloth to be used for the 
Thanksgiving dinner. 



82 



MATHEMATICAL CONSTRUCTION 



If 4 people can sit on one side, how many can sit at 2 
sides? (Ans. Two 4 people.) How many 4 sides? (Ans. 
Four 4 people.) 

Bisect the square, using a 4-inch ruler. Lay it on the top 
edge from upper left corner, drawing line at the end. Eepeat 



Beach 



1 

1 




1 m 

1 


1 


r 

Ni/ 


1 
1 

r ~7 

i ' 

i 



. p 
T"3 ' 



on lower edge, and draw bisecting line. 
Fold on this line. 

Now if the oblong is the table and 
there are 4 plates on the short end, how 
many on the long one ? (Ans. 2, 4 or 
8 plates.) (Some will say 8 plates on 
account of ruler measurement.) 

That is enough ruler work for this 
early ruler lesson. 



Fold one long free edge to meet fold. 

If small oblong thus formed is cloth enough for a Pil- 
grim's dress, how many dresses can she get from opposite 
large oblong? (Ans. 3 dresses.) 

Fold other long edge to center fold. Open. Fold the 
other way, making 8 small oblongs out of the 8 inch square. 
During process in giving problems call surfaces cloth for 
aprons instead. Open. 

Fold so that point b touches center a. 

Fold so that point c touches center a. 

Compare triangles thus formed (e and /), calling them 
Pilgrims' shawls. If it took $1.00 worth of yarn to make e, 
how much did it take to make f? etc. 

Some Indians had a wigwam here (pointing to d). Some 
of them ran \ mile along here (pointing to edge d-g-h) to 
the beach to see the Mayflower. The others ran along here 



MEASURING: SINGLE UNIT RULERS 



83 



(pointing to edge d-k-m) to the beach. How far did the 
others run? (Ans. Just as far, ■J mile.) 



Fold so that point d touches a. Cut out triangles 1, 2, 
Compare size of triangles, which form sails. 

If large triangle shows number of 
people who came over in Mayflower, and Ti 
small one shows how many got sick, what ! 
part of them got sick? 

Many more questions can be asked if 
there is time ; questions about clearing the 
ground, planting corn, chopping down 
trees, building log-cabins, etc. 



3,4. 




T.«3 2. 



LESSON III 
Cup 



Material : A sheet of manila paper 6"x9". 

Eulers: 6"3". 

Pass the 6-inch rulers saying, "This is a 6-inch 
ruler." 

Pass the 3-inch rulers saying, "This is a 3-inch 
ruler." 

Let the children put the rulers next to each other 

(Kg- 1). 

How many 3-inch rulers can be cut from the 

p.ji 6-inch ruler f 



84 MATHEMATICAL CONSTEUCTION 

Show me the part of the 6-inch ruler which equals 3" ; or, 
show me the part of the 6-inch ruler which is just as much as 
the 3-inch ruler. (Children show a.) 

Now show me the difference between the 6-inch ruler and 
the 3-inch ruler. (Children show b.) 

Show me the sum of 6" and 3". Children show 
Fig. 2. 

When in a later lesson they use rulers 3"6"9", 
they will be able to say that the sum of 3" and 6" 
equals 9". In this lesson it is enough to have them 
get an idea of what the word "sum" means. 

Let them estimate the length of the paper which 

they have. They will tell you that it is the sum of 

6" and 3" long, or 3'3" long, or 6" and J of 6" long. 

In estimating width, they will say: "The paper 

is 6" wide," or "The paper is 2'3" wide." 

Now teacher lays her 6-inch ruler on upper edge 
Ma- and makes a mark at a (Fig. 3). Children imitate. 

Eepeat on lower edge, marking at b. 
Draw line a-b. Cut on line. Compare sur- 
faces c and d. 

Play c is a towel. How many can be 
made from df If c costs a nickel, d costs 2 
nickels. If d costs a $1.00, c costs J of a f/?.3. 

dollar. 

c Now using 3-inch ruler, bisect all the 
edges. Do not draw the bisecting lines. 
They are not necessary. 

Fold this 6-inch square on one diagonal 
with bisecting marks on the outside (Fig. 4). 
Make point a touch mark b. 






MEASUEING 35 

Make point c touch point d. 

Turn back in opposite directions 
the two free points at e. Result : A drinking cup 
for use on a train, or in the park. 



LESSON IV 
Bookmark 

Material : A pretty thin cardboard, a piece of cord which 
harmonizes in color with the cardboard. 
Rulers: 2"4". 

Since cardboard comes in large sheets and must be cut up 
by the teacher, she may as well cut it the proper dimensions, 
4"x2", to save material. But let the children have the two 
rulers and compare them as in previous lesson, showing sum, 
difference, etc. Let them estimate length and width of paper. 
Let them see that sometimes the paper is just wide enough 
and does not need marking. 

Lay 2-inch ruler on upper and lower 
long edges respectively. Make marks, but do 
not draw bisecting line. It is not necessary. 
D Compare triangle thus formed to oppo- 

site surface. If it is enough leather for the 



B 



corner of a book, the opposite surface is enough for how 
many corners? If it costs a dime, how much does opposite 
surface cost? (3 dimes.) 

Fold so that point d touches a. Compare triangle thus 
formed to opposite large triangle. Relation J. Punch holes 



8C> 



MATHEMATICAL CONSTEUCTION 



on the free edges of the small triangles along line a-b. 
middle holes first equidistant from 
points a and b. Then the others equi- 
distant from center and points a and b, 
respectively. Lace these and you have 
a book mark which fits on the corner 
of the page. 



Punch 




LESSON V 



Stamp Pocket 



Get 



Material : A heavy, pretty paper. 

Eulers: 2"3"5". 

Make a 5-inch square, using a 5-inch ruler. 

Then give the 2-inch and 3-inch rulers to children, 
sum 2+3=5. 

Lay 3-inch ruler on upper edge from 
upper left-hand corner and make mark 
a. (Fig. 1). 

Eepeat on lower edge and make 
mark b. 

Draw the line. 

Lay 2-inch ruler on upper edge from 
upper left-hand corner and make mark 

c. Eepeat on lower edge, making mark 

d. Draw line. 

Lay 3-inch ruler along left edge from upper left corner 
and make mark e. Eepeat on right edge, making mark /. 
Draw the line e-f. 



C A 

G- \ I -... 

€ -! \ 






MEASURING: SINGLE UNIT RULERS 



87 



Lay 2 -inch ruler along left edge from upper left corner 
and make mark g. Repeat on right edge and make mark h. 
Draw the line. 

Fold so that lower edge touches 
upper edge. Open. (Fig. 2). 

Fold so that left edge touches line 
c-d. Open. 

Fold so that right edge touches line 
a-b. Open. 

Cut from e to fold. 

Cut from / to fold. 

Fold so that point g touches h. 

Fold so that point i touches h. 

Fold so that d-b line touches h-h line. 

Fold so that upper left corner touches point I. 

Fold so that upper right corner touches point m. 

Fold so that c-a line touches l-m line. 



i c 




*l • 














1 H 




K ; 



FI9.2. 




FI3.3 



Fold so that line a (Fig. 3) 
touches line b. 

Fold so that line d (Fig. 3) 
touches line c. 

Eesult : Fig. 4. 






F<3- f 



Fold under the little triangles 1, 2, 3, 
4 and paste. Result: A stamp holder 
with two pockets. Close it by making 
lower edge touch upper one. 



88 



MATHEMATICAL CONSTRUCTION 



LESSON VI 

Traveler's Pocket for Wash Cloth 



Material: Oilcloth (a 9-inch square) and a piece of tape 
14" long. 

Rulers 1" 7" 3" 4". 

Handling these rulers the children find that: 
3 4 3 4 

13 3 1 



The children have handled a 2-inch ruler in Lessons IV 
and V. They see a 2-inch length in the making of this wash- 
cloth pocket. So, even though they haven't the 2-inch length 
in their hands some will see : 









A 




c 




2 


t 

i; 


«r 


;l 


4 
2 
1 


3 

2 

2 


9 








s: 


<o 


j? 


- 


~ 


B 


--i 





.._j.« 


7 


7 


3 


3\ 


7 





With the 1-inch 
ruler mark off 1 inch 
on upper edge from 
upper left corner. Re- 
peat on lower edge and 
draw line a-b. 



MEASURING: SINGLE UNIT RULERS 89 

Lay 7-inch ruler on upper edge from point a and make 
mark c. Eepeat on lower edge and make mark d. Draw line. 

Lay 3-inch ruler along left edge from lower left corner 
and make mark e. Eepeat on right edge and make mark /. 
Draw line e-f. Lay 4-inch ruler on left edge, beginning at 
point e; mark point g. Eepeat on right edge and mark point 
h. Draw line g-h. 

Cut out oblongs 1, 2, 3, 4. 

Compare oblongs 5 and 6 (-J). 

Compare oblongs 6 and 8 (7). 

Compare oblongs 1 and 8 (J). 

Compare sum of 5 and 6 to 7. (They equal 2-7s.) 

Fold over the flaps 8 and 9. 

Fold 7 on top of them. 

Fasten with brass fasteners. 



Cut off triangles 1 and 2. 
Sew middle of tape to point t. 
Fold on dotted line and tie the tape on 
opposite side. 



2 



2 



r»$. x 



Q - 



b 6 

ft ji • 



LESSON VII 



Pilgrim's Bonnet 



Material: White paper cambric 
(a 6-inch square). 



"«" 



Eulers : 2"4"6 
Euler questions : 



90 



MATHEMATICAL CONSTRUCTION 



Put the 2-inch ruler next to the 
4-inch ruler. 

How many 2-inch rulers can be 
cut from the 4-inch ruler? 

Show part of 4-inch ruler which 
equals 2-inch ruler. 



/ 2 

Fig. 2. 

Show difference between 4-inch ruler and 2-inch ruler. 

Show sum of 2 inches and 4 inches. 
Lay 6-inch ruler next to this sum. 
How much is the sum ? 
Teacher writes on board : 2 



How many 2" in 6" ? 

Show difference between 6" and 4". 

In estimating length of 6-inch square, children will say: 

It is 6" long. 

It is the sum of 4" and 2" long. 

It is 3'2" long. 

It is the sum of 4" and -\ of 4" long, etc. 

Proceed with drawing of measurements. 

(Kg- 1.) 

Lay 4-inch ruler an upper and lower 
edges from left-hand side. Make marks and 
draw line a-b with 6-inch ruler. Use 2-inch 
ruler in same way, making line c-d. 

Laying 4-inch ruler along right and left edges from the 
top, make marks and draw line e-f. 

Laying 2-inch ruler in same way draw line g-h. Fold top 
edge to line g-h. 




MEASURING 91 

Compare this surface to opposite one (relation 1 to 5). 

If this cloth (pointing to small oblong) costs a nickel, 
what does this cost (pointing to opposite) ? (Ans. 5 nickels). 

If it costs a dime? (5 dimes. ) 

If it costs 2 cents? (5 2 cents.) 

Cut off this narrow oblong. 

Now fold back Jc-l-m on line g-h. Eesult: Fig. 2. 

Make cats 1, 2. 

Paste or sew squares x, y, z on top of one another. 

Cut into two equal strips the long, narrow oblong, which 
was cut from the 6-inch square. These are the ties. Sew 
or paste them on. 



LESSON VIII 

Stove 

Rulers: 3"6"9". 

Give grey school paper which comes 9"xl2". Ask ques- 
tions on rulers, bringing out following number relations : 

3 2 



9 


9 


6 




3 6 


2'3"=6" 


3)9 


3)6 


-6 


-3 


-3 




3 3 


3'3"=9" 






3 


T 


T 




T V 








Let the childr 


en 


estimate length and wi 


idth of 


paper. 


The } 


r will 


say 


: 










The 


papei 


■ is 


2' 


3" long. 
6" long. 














9"+3" long. 









92 



MATHEMATICAL CONSTRUCTION 



6"+3"+3" long. 
9"+J of 6" long, etc: 
3' 3" wide. 
9" wide. 
6"+3" wide. 
6"-fJ of 6" wide. 
With 9-inch ruler mark off 9 inches on upper and lower 
edges. Draw line and cut. Compare oblong to 9-inch 
square. Relation 1 to 3. 

If oblong shows cost of 1 lb. of meat, what does square 
show? (Ans. Cost of 3 lbs.) 

If it shows cost of 1 pint of milk, what does square show ? 
(Ans. Cost of 3 pints.) 

If 1 pint costs 3c, what does other cost? (Ans. 3' 3c 
or 9c.) 

The children ought to be able to infer 9c because they 
found out that 3'3" equals 9". 

Using 6-inch ruler for mark- 
ing and 9-inch ruler for drawing 
the line, make line a-b. 

Make marks c-d with 3-inch 
ruler. 

Draw line c-d with 9-inch 
ruler. 

Make marks e-f with 6-inch 
ruler (laying it on left and right 
edges from top). 
Draw line with 9-inch ruler. 
Make marks g-h with 3-inch ruler. 
Draw line with 9-inch ruler. 
Cut as indicated (Fig. 1) at g, h, e, f. 



1 

« 1 


N 


L i 





M ! 


P 

i . 



MEASUEING 



93 



Paste h and m under I. 

Paste n and p under o. 

Cut out free-hand, cutting so that the 
stove will have short legs. 

Make oven (a-Fig. 2), cutting 3 sides 
of it free-hand. Hinge is dotted line. 

Cut lids from top of stove. Make stove 
pipe out of oblong, which was cut from the 
9-inch square. 

Give 2-inch rulers to children. (The oblong was 9"x3"). 

Lay 2-inch ruler on right and left long edges, making 
marks. Draw line and cut. Result: Oblong 3"x2". Paste 
3-inch edges together, making cylinder for stove-pipe. Insert. 
Stove is made. 




LESSON IX 
Bank 

Pulers : 4"8"12". 

Material : Two 12-inch squares of manila paper cut from 
the school paper which comes 12"xl8". 

Use 6"xl2" pieces, which are 
left in Lesson X. 

Euler questions: 

How many 4" in 8" ? 

What part of 8" equals 4" ? 
Q of 8"). 

What is the difference between 
8" and 4"? 

What is the sum of 8" and 
4"? 



o A 

<x \ I 

t 

E r : 



6 



94 MATHEMATICAL CONSTRUCTION 

2'4" equals how many inches? 

3'4" equals how many inches? 

What part of 8" equals 4"? 

What part of 12" equals 4"? 

Estimate length and width of paper. 

It is 12" long. 

It is 3'4" long. 

It is the sum of 8" and 4" long. 

It is the sum of 8" and \ of 8" long. 

Lay 8-inch ruler on upper and lower edges, respectively, 
from left side. Mark a, b and draw line a-b. To draw line 
c-d use 4-inch ruler in same manner. 

To draw line e-f lay 8-inch ruler on left and right edges, 
respectively, from top of paper. Make marks e, f and draw 
line. To draw line g-h use 4-inch ruler in same manner. 

Fold on line e-f. Compare the two surfaces thus formed. 

If I saved $1.00 in so much time (pointing to small sur- 
face), how much should I save in so much time (pointing to 
large surface) ? If I save $4.00? (Ans. 2 $4.00, or $8.00— 
inference from ruler measurement). 

If a man earns $8.00 in so much time (indicating large 
surface), how much would he earn in so much time? (Ans. 
-I of $8.00 or $4.00.) 

If a man earns $8.00 and spends ^ of his money, how 
much does he spend ? ■ ...... 

Make cuts indicated at d, f, a, g. Fold and paste into box 
shape. 

Next day make another just like it, slipping it over the 
first like the cover of a box. Make a coin slit in the top. 
Bank is made. For decoration, see Measuring and Weaving, 
Lesson I, page 68. 



MEASUEING 



95 



LESSON X 
Envelope 

(In which to keep little freely cut units for decoration). 

Eulers: 5"1"7"2". 

Material : Paper left from previous lesson 6"xl2". 

In estimating length of paper, children will get sum of 
7" and 5" equals 12" even though the 12" ruler is not in 
their hands, for they have handled the 12-inch ruler twice 
while making the bank. 

In estimating width, they will get sum of 5" and 1" 
equals 6" for same reason. They have used 6-inch ruler in 
Lessons III, VII, and VIII. 

Other numerical values seen : 

5 7 7 5 6'2"=12" 

2-2-5 5 

7 5 2 2 



': 5 > 2 
.»• • .. . . . . ...«.., 

*! I 7 

i 

i ! 



12 

Lay 7-inch ruler on left and right 
edges, respectively, from upper edge; 
make marks and draw line a-b. 

Use 2-inch ruler in the same way. 
Draw line c-d. 

With 1-inch ruler make marks, e-f-g-h. 

Fold line e-f. 

Fold line g-h. (The children have 
no ruler at their desks long enough to 
draw this line.) 

Cut out oblongs 1, 2, 3, 4. 



96 



MATHEMATICAL CONSTEUCTION 



Fold back oblongs 6, 7, and 5. 

Compare 5 to opposite surface. Eelation 1 to 5. Play 
oblong 5 is blotter. 

How many blotters just as large can be cut from opposite 
surface ? 

If large blotting paper costs a nickel, what does small one 
cost? (1/5 of a nickel or la), eta 

Put paste on flaps 6 and 7. 

Paste 8 on them. 

Cutting from c to e and from d to g makes the fold over 
flap look better. 



LESSON XI 

Taboret 

Enters: 8"5"3"1". 

Material : An 8-inch square of rather heavy paper. 

New numerical relations seen : 



8 
-3 



8 
-5 



8 5 3 

Lay 5-inch ruler on upper and lower edges, respectively, 
from left edge, making marks a, b. c 

Draw line a-b. Lay 3-inch ruler in 
same way and draw line c-d. 

Lay 5-inch ruler on left and 
right edges, respectively, from upper 
edge ; make marks e-f and draw line. 

Lay 3-inch ruler in same man- 
ner and draw line g-li. Make the 



3i 



MEASUEING 



97 



marks which are necessary to draw the dotted edges of the 
^^^~>. small oblongs 1, 2, 3, 4, with a 1-inch ruler. 

^^V^J^J Draw the lines with the 3-inch ruler. 
i Cut out oblongs 1, 2, 3, 4. 

Make cuts indicated at marks e, f, g, h. 
Paste into box shape. (Fig. 2.) 
Cut out sides in any desired way to form 
legs. 



LESSON XII 
Chair for Father Bear 
(Story of Three Bears). 
Rulers: 3"6"9"12". 

Material: School paper 9"xl2". Numerical relations 
seen in comparisons of rulers and estimation of length and 
width of paper. 











4 


2 


2'3"= 6" 




3" 


are J of 6" 


3)12" 


6)12 


3'3"= 9" 




3" 


are ^ of 9" 


3 


2 


4'3"=12" 




3" 


are J of 12" 


3)"~9~ 


3)~6 


2'6"=12" 












3 3 


6 


6 


3 6 9 9 


12 12 


12 


6 9 


6 


3 


3 -3 -3 -6 


-9 -6 


-3 


9 12 


12 


3 
12 


3 3 6 3 
9 


3 6 


9 



98 



MATHEMATICAL CONSTRUCTION 



7 I ; »|* 

* j * i : 
-• -i i . 

/ ' 2 3 5 



G -- 



Lay 9-inch ruler on upper and 
lower long edges respectively, from 
left edge. Make marks a, b. Draw 
line a-b. Lay 6-inch ruler in same 
way and draw line c-d. Lay 3- 
inch ruler same way and draw line 
e-f. F u e 

Lay 6-inch ruler on left and right short edges, respec- 
tively, make marks g-h and draw line g-h. 

Lay 3-inch ruler in same way and draw line i-j. 

Make cut at b. 

Fold back surface 1-2-3. Compare to opposite surface. 
(Relation 1 to 3.) 

If it takes a roll of wall paper for this wall (indicating 
small surface), how many for this? (3 rolls.) 

If it takes 3 rolls for small wall, how many for large 
wall (3'3 rolls or 9 rolls-inference from rulers). 

Make cuts indicated at d and a. 

Fold back square 4. 

Now compare surface 1-2 to opposite surface. Eelation 
1 to 4. 

If wall paper for small surface costs $3.00, paper for large 



pji 



"00 



wall costs how much? (4 r $3.00 or $12.00- 
inference from ruler measurement.) 

If it took 12 pails of plaster for large wall, 
how many for small wall? (J of 12 pails or 3 
pails), etc. 

Make cut indicated at i. 

Fold and paste so that square 5 is under 3 ; 
4 under 6 ; and 7 under 3. 



MEASUEING: SINGLE UNIT RULERS 99 

Fold and paste surface 1-2 under 8-9. 

Cut out sides to form legs in any desired design. 

LESSON XIII 
Mother Bear's Chair 

Eulers: 2"4"6"8". 
Material : Stiff paper, 6"x8". 

In comparing rulers and estimating length and width 
of paper, these numerical relations are seen : 









4 


2 


3 


2 , 2 , '=4 ,/ 




2" are J of 4" 


2)T 


4)T 


2)6 


3'2"=6" 




2" are J of 6" 


2 






4'2"=8" 




2" are \ of 8" 


2)4 






2'4"=8" 




4" are J of 8" 








2 


2 


2 4 8 8 


8 6 


6 4 




2 


4 


6 4-6-4 


-2 -4 


-2 -2 





4688246242 
Use rules 2"4"6" to divide surface into 12 squares, as the 
3"6"9" rulers were used in previous lessons. 

Make cuts indicated at a and b. 
Fold back surface 1-2. Compare 
3 it to opposite surface. Relation 1 
to 5. 

Make cut indicated at c. 
Fold back surface 3-4. 
Compare this to opposite surface. 
Relation 1 to 4. 



7 



— 1 — r~ 



10 



M 



100 MATHEMATICAL CONSTKUCTION 

If this floor (showing surface 3-4) requires 2 square yards 
of carpet to cover it; what does this floor require? (showing 
large surface.) (Ans. 4'2 square yards or 8 square yards- 
inference from rulers). 

Make cut indicated at d. 

Fold and paste so that square I is under 5 ; 6 is under 2 ; 
3 is under 5. 

Fold back and paste 7-8 on 9-10 to strengthen back of 
chair. 

Make legs and back same design as Father Bear's Chair. 



LESSON XIV 
Baby Bear's Chair 
Rulers : 1"2"3"4". 



8 ^ i Material: A 4-inch square of stiff paper. 

Divide this into 16 1-inch squares by using rulers 1"2"3". 

Cut off one row of squares. 

Play both surfaces are rugs with fringe on short edges. 
(Fig. 1.) 

If 1 yd. of fringe is on one short edge of rag B, how 
many yds. are on one short edge of rug A ? 

How many yds. on both short ends of a? (2'3 yds., or 
6 yds.-inference from former ruler measurement.) 

How many yds. on both short ends of rug B? (2 yds.) 

How many times as much fringe must we have for the 
big rug? (3 times as much.) 

3'2 yds. are how many yards? (Inference from ruler 
measurement in previous lessons.) 



MEASURING: SINGLE UNIT RULERS 



101 



D A 




Make cuts indicated at a, b, c, d. 

Fold and paste chair as in previous 
lessons. Make legs and back same design 
as father and mother bears' chairs. 

Number relations seen: 



2'1"=2" 1" is i of 2" 
3'1"=3" 1" is i of 3" 



1112 
12 3 2 



2 
2)T 



4'1"=4" 1" is J of 4" 
2'2"=4" 2" are J of 4" 



4 4 4 
-3 -2 -1 



LESSON XV 

Father Bear's Bed — Bulers :3" 6" 9" 12". 

Material : 2 sheets stiff paper, 9"xl2". 

Divide paper into 12 small squares, using 
rulers 3"6"9", as in Lesson XII. 

Make cuts indicated at a, &. 
squares 1-2. Compare this surface to opposite 
large surface. (Belation 1 to 5). If it is 
material for 1 sheet, how many sheets can 
be made from large surface (5 sheets). Fold 
back 3. Make cut indicated at c. Fold back 4, 5, 6. Com- 




102 



MATHEMATICAL CONSTEUCTION 



pare 4, 5, 6 to opposite surface. Eelation 1 to 2. If this is 
material for a pillow-case, how many can be made from large 
surface? (Ans. 2 pillow-cases.) 

If it takes 3 minutes to hem one pillow-case, how long 
will it take to hem 2? (Ans. 2'3 minutes, or 6 minutes.) 
Make cut d. Fold and paste into box shape. 

This is enough for one lesson. Next day take the other 
9"xl2" sheet of paper and make another box shape exactly 
the same. Compare volumes of these two. If one holds 
3 lbs. of feathers, the other holds 3 lbs. of feathers. 

Now cut one box shape so that it will hold half the 
amount of feathers. Eye measurement only. 

Now if small box holds 3 lbs. of feathers, large one holds 
6 lbs. If large one holds 12 lbs., small one holds J of 12 lbs., 
or 6 lbs. 



Ftf.fc. 



From small box shape cut 

so that the legs of the bed are 

formed. 

From large box shape cut 

half-way down on the edges 

a and b. Cut across on line c. 

This is the foot of the bed. 

Leave opposite end as it is. 

This is the head of the bed. 
Now to get the sides, cut down half-way again on what is left 
of edges a and b. Then cut across to head of bed. 

Paste Fig. 3 on Fig. 2 and bed 
is constructed. Numerical rela- 
tions found in Lesson XII are 
reviewed. 




Hr 



MEASURING: SINGLE UNIT RULERS 103 

LESSON XVI 

Mother Bear's Bei>— Eulers : 2"4"6"8". 

Materials : 2 sheets stiff paper 6"x8". 
Make just as father's bed was made, but use rulers 
g"4"6"g" instead. Numerical relations found in Lesson 

XIII, reviewed. 

LESSON XVII 

Baby Bear's Bed— Rulers: 1"2"3"4". 

Material : 2 sheets stiff paper 3"x4". 
Make the same as Father's and Mother's Bed, but use 
rulers 1"2"3"4". Numerical relations found in Lesson 

XIV, reviewed. 

LESSON XVIII 

Bowl for Father Bear — Eulers : 12"4"3"1". 

Material : Piece of paper 12"x4" and a 4-inch square. 



Numerical relations seen: 






4 


3 






3)12" 


4)12" 




3'4"=12" 
4'3"=12" 




3 4 


4 






1 -1 


-3 





104 MATHEMATICAL CONSTRUCTION 



On left and right edges, 
measuring from upper edge 

a [ | ,ujjU(J1U1.1./ _ .c. - .UI'utMimJ a with 3_inch rulei \ make 

marks a, b. Draw line a-b. 

Fold so that lower edge of 

paper touches line a-b. 

Cut on the fold. Compare this long narrow oblong to 
large surface. Relation 1 to 7. Now fold on line a-b. Then 
on the long narrow oblong c, make vertical cuts about J of 
an inch apart up to the line a-b. 

Paste right edge on left edge to* form cylinder, seeing that 
the little cuts on the bottom turn inward. These are the flaps. 
Put paste on the bottom of the flaps. Lay this cylinder on 
the four-inch square with flaps down. When the flaps are 
pasted to the square cut off the parts of the square which 
protrude. Bowl 3 inches high is made. 



LESSON XIX 

Mother Bear's Bowl— Rulers: 12"3"2"1". 

Material : Piece of paper 12"x3" ; a 4-inch square. 

Number relations seen in this lesson : 

4 

— 2 3 3 
1 -2 -1 



6'2"=12" 3 ) 12 



4' 3"=12" _^_ 

12'1"=12" 2)12 3 1 



MEASURING: SINGLE UNIT EULEES 105 



Lay 2 -inch ruler on left 
and right edges respectively. 
Make marks a, b. Draw line 
with 12-inch ruler. Fold so 
that lower edge touches line 



UJIdHJIULUi C.JLJUlLliJL'l'i 



a-b. Cut on fold. Compare this to large surface. Eelation 
1 to 5. Fold on line a-b. Compare this long narrow oblong 
to opposite surface. Eelation 1 to 4. If it is a board in the 
floor what is the other? (4 boards). 

If it costs 3c, what does other cost? 4'3c or 12c. 

Make the little cuts for flaps up to line a-b. 

Paste right on left edge, seeing that flaps turn inward. 
Put paste on bottom of flaps. Set cylinder so that flaps have 
their paste side down on the 4-inch square. Cut away pro- 
truding surface of square. Mother Bear's Bowl 2" high is 
made. 

LESSON XX 
Baby Bear's Bowl— Rulers: 12"2"1". 

Material : Strip of paper 12"x2" ; a 4-inch square. 

On left and right edges 
measuring from upper edge 
make marks a, b with 1-inch 
ruler. Draw line a-b with ^ 
12-inch ruler. Fold so that 
bottom edge touches line 

a-b. Cut on fold. Compare the oblong cut off to large 
surface. Relation 1 to 3. Fold on line a-b. Make vertical 
cuts up to line a-b on oblong c for flaps. Paste right edge 
on left edge, turning flaps inward. Put paste on bottom of 



mwimr. .g. . imimuim' 



e 



Q 



So 



106 MATHEMATICAL CONSTRUCTION 

them, set on 4-inch square and cut off protruding part of 
square. Baby Bear's Bowl is 1" high. 

If baby's bowl contains a 

pint of porridge, what does 

papa's contain (3 pints) ? 

What does mamma's contain 

(2 pints) ? The child cannot 

say 1 quart, unless you have 

had the actual quart and pint measurements in your room 

and he has measured and found out that a quart equals two 

pints. 

The child should deal with the actual measures first. 
Then after he has seen the relations, let him draw infer- 
ences, by applying those relations in other ways. 

For instance: When we made a bowl for the baby bear, 
and another with twice its capacity for the mother bear, we did 
not make them with the actual capacity of a pint and quart. 
But we may assume that the baby's holds a pint. Then the 
child infers that the mother's (having twice the capacity) 
holds a quart, because he has previously discovered through 
actual measurement that a quart is twice as much as a pint. 



LESSON XXI 

Pail for Jack— Eulebs : 9"5"4"1". 

Material: Pretty colored paper 9"x5"; a 3-inch square. 
Number relations seen: 



MEASURING: SINGLE UNIT BULEES 



107 



5 4 4 
4 14 



5 5 
-4 -1 



9 5 1 



Mld'JUL.c --UwlUWil 



On left and right edges, meas- 
uring from upper edge with 4-inch 
ruler make marks a, b. Draw line 
a-b. Fold so that lower edge 
touches line a-b. Cut on fold. 
Now fold on line a-b. Make little 
vertical cuts on oblong c free-hand about J of an inch apart 
up to line a-b. Paste right edge on left one, flaps turning 
in. Put paste on bottom of flaps, lay on 3-inch square and 
cut away part of square which is left on the outside. Use 
first strip x which was cut off as a handle for Jack's pail. 



LESSON XXII 

Jill's Pail— Rulers : 9"3"2"1". 

Material : Pretty colored paper 9"x3" ; a 3-inch square. 

Lay 2-inch ruler on left and 
right edges respectively from up- 
per edge. 

Make marks a, b and draw 
line a-b. Fold so that lower edge touches line a-b. Cut on 
fold. Use this strip x for handle. Fold on line a-b. Cut c 
into flaps. Paste right edge on- left one, turning flaps in. 
Put paste on bottom of flaps and paste the cylinder on to the 



a ]iLU11JM1 3 JUUiDUlf 



108 MATHEMATICAL CONSTRUCTION 

3-inch square. Cut off protruding parts of square. Paste 
on the handle. Jack's pail is 4" high. 
•s JilPs pail is 2" high. Compare vol- 

umes. If Jill's holds a quart of water, 
F-L—^J Jack's holds 2 quarts. 

If Jack's holds a gallon of water, 
Jill's holds J of a gallon. If Jill's 
holds a pint of water, Jack's holds 2 
pints or 1 quart (if children have had 
pint and quart measurements). 

If a pint costs 3c, what does a quart cost? (2' 3c or 
6c) etc. 




LESSON XXIII 

Fox's Dish— Eulers: 5"2"1". 

Material : Paper 5"x2" ; a 2-inch square. 

Using 1-inch ruler make marks 
a, b. Draw line a-b with 5-inch 
ruler. Fold so that lower edge 
touches line a-b. 

Cut on fold. Use pieces of this 
strip to make side handles for dish 
when it is finished. 

Fold on line a-b. Cut c into flaps. Paste right on left 
edges, flaps turning in. Put paste on bottom of flaps. Lay 



WMUJ. Mil MM 



MEASURING: SINGLE UNIT RULERS 



109 



cylinder flaps downward on 2-inch square. Cut off parts of 
square which project beyond cylinder. 

LESSON XXIV 

Stork's Dish— Eulers : 5"7"6"1". 

Material: Paper 5"x7"; a 2-inch square. 
Number relations seen : 
5 6 6 7 6 7 
11-1-1 -5 -6 




< 



) 



6 7 5 6 11 

Lay 6-inch ruler on left and right edges 

respectively. Make marks a, b. Draw line 

a-b. Fold so that lower edge touches line 

a-b. Proceed as in previous lesson to make 

the Stork's dish. Use strip cut off for side handles. 

Relation of Fox's dish to Stork's dish 

is 1 to 6. If Fox's dish contains lc worth 

of meat, what does Stork's contain? (6c 

worth.) 

If Fox's dish contains 2c worth, what 
does Stork's contain? (6' 2c worth or 
12c worth.) 

LESSON XXV 

Handbag— Eulers : 9"6"3"2"1". 

Material : Paper 9"6" ; also a strip 9"xl". 
Number relations reviewed — a great many seen, in com- 
paring rulers and estimating length and width of paper. 



TJ 



110 



MATHEMATICAL CONSTRUCTION 



9+3=6+6 



6 2 3 
3 12 

9 3 1 

6 



2'3"=6" _* c 

3'2"=6" 

3'3"=9" 



£ 1 



H D 



F J 



Lay 3-inch ruler on left and right 
edges respectively. Make marks a, b; 
ff]\ draw line a-b with 9-inch ruler. 

' f -^» Lay 2-inch ruler on upper and lower 

edges, respectively, measuring from left 
edge make marks c, d, and draw the line. 
Lay 2-inch ruler on upper and lower 
edges respectively, measuring from right 
edge; make marks e, f and draw line. 

Lay 1-inch ruler on upper and lower edges respectively, 
measuring from left side; make marks g, li, and then draw 
line g-h. 

Doing same on right side make marks i, j, and then draw 
line i-j. 

Make cuts indicated at a and b. Fold on lines c-d and 
e-f. 

Now fold back on lines g-h and *-/. Fold on line a-b, 
so that all these small folds are inside. 
Paste oblongs 1 and 2 together. 
Paste oblongs 3 and 4 together. 

Cut the 9"xl" strips into two equal narrow strips for 
handles — eye judgment only. 



MEASUEING: SINGLE UNIT KULEKS 



111 



LESSON XXVI 
Pencil-box with Lid — Eulers: 10"2"3"5". 

Material : Thin, smooth cardboard 10"x6". 
Number relations seen : 

2 5 3 2'3"= 6" 

3 5 3 3'2"= 6" 
- _ 5'2"=10" 
5 10 2 

2 



10 




Lay 5-inch ruler on left and right edges, measuring from 
upper edges ; make marks a, b and draw the line. 

Lay 3-inch ruler in same manner; make marks c, d, and 
draw the line. 

Lay 2-inch ruler in same manner; make marks e, f, and 
draw the line. 

Make marks g, h, with 1-inch ruler; draw line with long 
ruler. 

Make marks i, ; with 1-inch ruler; draw line with long 
ruler. 

Cut out oblongs x and y. 

Make cuts indicated at a, b, c, d. 

Fold up and paste. 

Surfaces can be compared by giving concrete problems 
as in former lessons. 



112 



MATHEMATICAL CONSTRUCTION 



LESSON XXVII 
Match-safe 

Colored smooth, thin cardboard 10"x4" for back. 
Colored smooth, thin cardboard 6"x5" for box. 
Sandpaper, 3"x2". 
Kulers: 5"4"3"2"1". 

Let children estimate 10" length of cardboard with these 
rulers. They will say : 

5 5 4 3 4 3 2 

4 3 3 5 4 3 2 

12 2 12 3 2 

10 10 1 1 10 1 2 



10 10 



10 



10 



C 6 



C A 

T 



On the cardboard which is 6"x5", lay 
5-inch ruler on upper and lower edges, 
L respectively, measuring from left edge; 
j make marks a, h, and draw line. Lay 
4-inch ruler in same way; make marks 
hp 6© c, d, and draw line. Lay 2-inch rule 
in same way; make marks e, f and draw line. Lay 1-inch 
ruler in same way ; make marks g, h and draw line. 



MEASUEING: SINGLE UNIT RULEES 



113 



Lay 3-inch ruler on left and right edges, 
respectively, measuring from upper edge ; make 
marks i, j and draw line. 

Lay 2-inch ruler in same way; make marks 
Jc, I and draw line. 

Cut out oblongs 1, 2, 3, 4. Make cuts indi- 
cated at i, j, k, I. Fold and paste into box 
shape. 

Paste box and sandpaper on large card- 
board — spacing agreeably. 




LESSON XXVIII 



Wood-box — Rulers 



2" 


9-" 


(o" 


r 










3 








<f' 
















9' \ 








n" \ 


i 







Material: Stiff paper 8"xl2". 
Using 2"4"6"8" rulers to divide the 
short edges into 4 equal parts, gives a 
good review of number relations found 
in Lessons XIII and XVI. 

Using 3"6 ,, 9' , 12 // rulers to divide 
long edges into 4 equal parts gives a 
review of number relations found in 
Lessons XII and XV. Make cuts in- 
dicated in Fig. 1. 



Fold and paste into box shape. (Fig. 2.) 
Cut on dotted lines a and b. In comparing 
surfaces and lines, speak of cost of coal or 
wood. Use the words "ton of coal," "cord of 
wood." They hear measurement words used in connection 
with things which require those measures. 



114 



MATHEMATICAL CONSTRUCTION 



LESSON XXIX 
Sled— Rulers : 5"4"3"2"1". 



Material: Paper 5"x3". 
Number combinations seen : 



3 2 
1 2 
1 1 





/'' 


**: . 










B 




A 

















2 3 4 3 
1112 

3 4 5 5 



5 5 etc. 

Make vertical dotted lines with 4-inch and 1-inch rulers, 
respectively, measuring from left edge. Make horizontal 
dotted lines with 2-inch and 1-inch rulers, respectively, meas- 
uring from the top. 

Cut out rectangles a and h. 

By folding back rectangles formed by dotted lines there 
are surfaces to compare which show the relation 1 to 2; 
1 to 3 ; 1 to 4. 

Play these rectangles are sidewalks covered with snow to 
be shoveled. If a boy can shovel this (pointing to small 
surface) in 1 hour, how many hours will it take him to 
shovel this (pointing to the opposite surface). Result de- 



MEASURING: SINGLE UNIT EULEES 



115 



pends on the surfaces which the teacher is comparing, of 
course. Suppose relation to be 1 to 2. (Ans. 2 hours.) 

If he shovels the small walk in 2 hours, he can shovel 
the large one 

in 2x2 hours, or 4 hours, 
in 2x3 hours, or 6 hours. 
in 2x4 hours, or 8 hours. 
in 2x5 hours, or 10 hours. 
in 2x6 hours, or 12 hours. 

Many children will be able to give these results on account 
of former experiences with single unit rulers. 



LESSOR XXX 



Pushcart— Eulers : 7"5"4"1"2". 



Materials: Stiff school paper 9"x6". 

Ask children how long and wide 
the paper is without using any ruler. 
They have handled the 9-inch and 
6-inch rulers so often that they ought 
to know how they look by this time. 
Now tell them to measure the 
length with the rulers which they 
have. They learn that the sum of 
the 7-inch and 2-inch rulers equal 9"; that the sum of the 
5-inch and 4-inch equal 9". 

In measuring the width they learn that 5" and 1" equals 
6"; that 4" and 2" equals 6". 




116 MATHEMATICAL CONSTKUCTION 

Cut on dotted line drawn between marks made on upper 
and lower long edges with 7-inch ruler. 

The rectangle a is now 2"x6". With the 2-inch and 
4-inch rulers divide this into 3 2-inch squares. 

Cutting off the corners of two of these squares in a 
curved line gives two circles to be uesd as wheels. ____ 
Cut remaining square into two equal pieces to be j > 

used as supports for the axles, which are tooth- ! . 

picks. »j2 

Now cut on dotted line, which is drawn be- 
tween marks made on right and left edges with 5-inch 
ruler. 

Cut rectangle b (Fig. 1) into two equal narrow strips 
to be used as shafts. From c (7"x5") the box shape is 
made. (Fig. 3.) 

With 1-inch ruler make marks a, b 
on upper and lower edges, measuring 
from left side. Draw line a-b. With 
1-inch ruler make marks c, d on upper 
and lower edges, measuring from right 
side. Draw line c-d. 



With 4-inch ruler, measuring from 
top on the left and right edges, make marks and draw line e-f. 

With 1-inch ruler, measuring from top, make marks on 
left and right edges and draw line g-h. 

Make cuts indicated at e, f, g, h. 

Fold and paste into box shape. 

Paste the little supports for the axles on the under side 
of the box after the axles have been laid in place. Put on 
the wheels and paste on the shafts. (Fig. 4.) 



MEASURING: SINGLE UNIT RULERS 



117 




In comparing lines and surfaces in 
this lesson, base concrete problems on 
roads, trees, houses, stores or parks; 
things which the banana-man sees when 
he pushes his cart through the streets. 

Ask about time or number of men required or material 
needed to pave or sprinkle roads; to cut or sprinkle grass 
in the park; to plant trees in a row. Ask about amount of 
houses of equal size which can be built on compared surfaces. 



LESSON XXXI 

Gocart— Eulers : 2"4"6"8". 

Material : 2 sheets of stiff paper, one 10"x8" ; the other 
8"x6". 

In estimating length of paper new number combinations 
seen are : 
8 6 
2 4 



10 10 



Some children ought to know 
that the paper is 10" long, for they 
talked about 10" in Lessons XXVI 
and XXVII. 

Measuring on upper and lower 
long edges from left side with the 
8-inch ruler on the large paper, 
make, marks and draw line a-b. 

Cut on dotted line a-b. Com- 
pare c to dg, with questions about 
material and cost of material for doll quilts and mattresses. 




118 MATHEMATICAL CONSTKUCTION 

From c measure and cut 4'2" squares, using the 6"4"2" 
rulers. From these squares cut free-hand the four wheels, 
keeping them 2" in diameter. 

Measuring on left and right edges A 

of 8" square from the upper edge with ' ' * 

the 6-inch ruler make marks, and draw A 

line e-f. Cut on line e-f. From g, U 

four equal long narrow strips are to 

be cut. Two of these, to be used for 

the handle, need a 2-inch mark at one -p, 3 ^_ 

end of each. 

Fold so that line a (Fig. 2) touches this mark. 

Place short-folded parts of these strips together and 
paste, forming handle. (Fig. 2.) 

Mark other two strips as indicated in Fig. 3. 

Fold on marks. Paste 
a on b. These are the 
springs to be pasted under 
■^S^— the box-shape when it is 

made, putting toothpicks 
through the inch square sides of the springs for axles. 

To make box-shape use d. (Fig. 
1.) It is 8"x6". c A 

Measuring with 6-inch ruler on | I T 

upper and lower long edges from left £...{ J. 

edge make marks and draw line a-b. 
In same way using 2-inch ruler draw 
line c-d. Measuring with 4-inch 




ruler, on left and right edges from r '3 h- 

upper edge, make marks and draw 

line e-f. In same way using 2-inch ruler draw line g-h. 



MEASTJBING: SINGLE UNIT EULEES 



119 



Make cuts indicated at a, h, c, d. Fold and paste into 
box-shape. 

This is enough for one lesson. Next day give other piece 
of paper 8"x6" to children. Make another box shape from 
it like Fig. 4. Compare volumes of these two which are 
equal. 

When mamma goes marketing she 
takes baby with her in the go-cart. If 
one volume holds a peck of potatoes, the 
other holds a peck of potatoes. 

If one holds half a peck of apples 

the other holds half a peck of apples. 

If one holds 5 pounds of sugar, the other 

holds 5 pounds, etc. 

Now stand last box up on end and inside of the first 

box which has the springs and axles under it. Put on the 

wheels and paste on the handle. 




LESSON XXXII 

Cradle— Rulers : 1"9"5"4". 

Materials : 2 sheets of stiff paper 10"xl6" 
stiff paper 12" square. 

Use one sheet of the paper, 10"x6", first. 

In estimating length, get new relations: 
9 5 
1 4 

10 1 



1 sheet of 



10 



120 



MATHEMATICAL CONSTKUCTION 




Ti?( 



Measure on upper and lower 
edges, from left edge with 9-inch 
ruler; make marks and draw line 
a-b. Eepeat with 1-inch ruler, and 
draw line c-d. Measure on left 
and right edges from upper edge 
with 5-inch ruler ; make marks and 
draw line e-f. Eepeat with 1-inch 
ruler and draw line g-h. Make 
cuts indicated at e, f, g, h. Fold 
and paste into box shape. 

Take other sheet, 10"x6". 
(Fig. 2.) 

Measure on upper and lower 
edges from left edge with 4-inch ruler; make marks and 
draw line a-b. Measure on upper and lower edges again with 

4-inch ruler, but measure from 
points a and b instead of left edge ; 
make marks and draw line c-d. 
Cut on dotted lines a-b and c-d. 
Compare rectangles, calling them 
rugs. Compare sides of rectangles 
by asking questions about length 
and price of fringe. 
Fold so that each rectangle is folded into 4 equal parts ; 
folding long edges so that they meet in each instance. The 
two larger rectangles e 
and / (they look like Fig. 
3 now) are to be fashion- 
ed into rockers, after they 
are pasted in place under 
the box-shape. 



E ; F C 



TV z 



Ti'j'b 







MEASURING: SINGLE UNIT EULEES 



121 



'?¥ 



Cut off the protruding flaps indicated at 1, 2, 3 and 4. 
Slope bottom edge of vertical surface to look like rockers. 
Cut £ into strips and use them as braces to 
keep rockers in place. 

Next day make another box shape out of 
the 12-inch square thus: 

On upper and lower edges, measuring 

from left edge with 4-inch ruler, make marks 

and draw line a-b (Fig. 4), using same ruler 

on same edges, but measuring from right edges draw line c-d. 

Same ruler on left and right edges measuring from upper 

edge, draw line e-f. 

Same ruler, same edges, measuring from lower edge, draw 
line g-h. 

Make cuts indicated at e, f, g, h. Fold and paste into 
box shape. Compare this volume 
to the one made yesterday. Rela- 
tion 2 to 1. If one holds a quart 
of beans the other holds ^ of a 
quart. 

If one quart costs 10c, ^ quart 
costs J of 10c, or 5c. 

If one quart costs 12c, -| quart costs -J of 12c, or 6c, etc. 
Set large volume on one end inside the other one. 
(Fig. 5.) 

LESSON XXXIII 
Bureau— Rulers : 3"6"9"12"— 1st Day 

Material: 1st day, 12-inch square of stiff paper. 

Divide 12-inch square into 16 small squares, using the 
3" 6" 9" rulers. Cut, fold and paste into box shape, having 
dimensions 6"x6"x3". 




■F, f f 



122 



EATHEMATICAL CONSTRUCTION 



Material : 2nd day, stiff paper 10"x7". 

Kulers : 8"2"5". 

Number relations seen : 8 5 

2 2 



•S / 



10 7 
Lay 8-inch ruler on upper 
and lower edges, measuring 
from left edge; make marks 
and draw line a-b. (Fig. 1.) ' 

Lay 2-inch ruler in same 
way ; draw line c-d. c 

Lay 2-inch ruler on left 
and right edges, measuring 
from upper edge; make marks 
and draw line e-f. Lay 2-inch ruler on same edges, but meas- 
ure from lower edge, and draw line g-li. 

Make cuts indicated at a, b, c, d. Fold into box shape; 
dimensions 6"x3"x2". Make two more 
boxes with dimensions 6"x3"x2". Com- 
pare their volumes to each other and to the 
large one made the first day. Eelation 1 to 
3. The small box shapes are drawers to be 
slid into large frame 6"x6"x3". If one 
drawer holds 3 tablecloths, how many will 
the whole bureau hold? (Ans. 3'3 table- 
cloths, or 9 tablecloths.) If the tablecloths 
in one drawer cost $4.00, how much does the bureau full 
cost? (Ans. 3'$4.00, or $12.00.) 

If they cost $2.00? (Ans. 3'$2.00, or $6.00.) 

Slide the drawers into the frame. Give the children an- 



r£ 



MEASURING: SINGLE UNIT EULEES 



123 



other small piece of the paper. From this let them make a 
back for the top free-hand; also the handles for the drawers. 
Silver paper is a very good play substitute for glass, if the 
children care to have a mirror at the top. Put on extra legs 
at bottom if desired, making them like tiny square prisms. 




LESSON XXXIV 

A Chiffonier 

Euleks: 10"2"8"; 2"4"6". 

1 sheet stiff paper 12"xl0". 
4 sheets stiff paper 8"x 6". 
1 sheet stiff paper 6"x 6". 
1 sheet stiff paper 10"x 8". 

Make line a-b with 10-inch ruler, 
measuring from top of paper, which 



is 12"xl0". (Fig. 1.) 

Make line c-d with 2-inch ruler, 
measuring from top. 

Make line e-f with 8-inch ruler, 
measuring from left. 

Make line g-h with 2-inch ruler, 
3' measuring from left. 

Make cuts indicated at e, f, g, h. Fold and paste into 
box shape. Large frame is made. (Fig. 1.) 

Make line a-b (Fig. 2, paper 
8"x6") with 6-inch ruler, measuring 
from left. Use 2-inch ruler same 
way to make line c-d. 

Use 4-inch ruler to make line e-f, 
measuring from top. 

Use 2-inch ruler same way to /r/ ^- 2 ' 

make line g-h. Make cuts indicated at e, f, g, h. 



Gr ! 


e- 


A: 


to • 

— • 




F 


1 
1 




B[ 



124 



MATHEMATICAL CONSTRUCTION 




Fold and paste into box shape. Make four of these. 
They are the four drawers to the right. From 6-inch square 
make a 2-inch cube with one side open, using 2- and 4-inch 
rulers. This is the little drawer for the upper left-hand 
corner. 

a „ Use 8-inch ruler to get line a-b, measuring 

from top. (Fig. 3.) Use 2-inch ruler same 
way to get line c-d. 

Use 6-inch ruler, measuring from left, to 
get line e-f. 

Use 4-inch ruler same way to get line g-h. 
Use 2-inch ruler same way to get line i-j. 
Cut out upper and lower left-hand corners. 
Make cuts indicated at e, f, g, h. 

Fold and paste into square prism shape. The side which 
has three free edges is the door of the cupboard. (Fig. 4.) 

Make legs and mirror back as in pre- 
vious lesson if desired. 

Compare volume of a and b (equal). 
Compare volume of e and a (1 to 2). 
Compare volume of e and f (1 to 3). 
Compare volume of e to sum of a and b 
(1 to 4). 

Compare volume of e to sum of f and d 
(1 to 5). 

Compare volume of / to whole frame before any drawers 
were put into it (1 to 4). 

Compare volume of a to volume of large frame (1 to 6). 

If e holds two of baby's dresses, a holds 2 times 2 dresses, 

or 4 dresses ; / holds 3 times 2 dresses, or 6 dresses ; the sum 

of a and b holds 4 times 2 dresses, or 8 dresses; the sum of 




MEASURING: SINGLE UNIT EULEES 



125 



a, &, and c, holds 6 times 2 dresses, or 12 dresses; the sum of 
a and / holds 5 times 2 dresses, or 10 dresses, etc. 

In this lesson the children handled the rulers 2",4",6",8", 
10", and one of the papers was 12" long. They have dealt 
with this length so often that they know it, even though the 
12-inch ruler is not in their hands. 

Thev saw that: 



6'2"=12" 
2'4"= 8" 
2'6"=12" 



2'2"= 4" 
3'2"= 6" 
4'2"= 8" 
5'2"=10" 

They also saw the combination, separation and division 
of these number facts. 

LESSON XXXV 
A Large Envelope 
(To hold weaving strips, or braids of cord or raffia until 
they are needed.) 

Killers: 11"1"8"4". 

Material: 2 sheets manila paper 12"x9". 
New numerical values seen: 11 8 

1 1 




126 



MATHEMATICAL CONSTEUCTION 



Make line a-b, measuring from the left with the 11-inch 
ruler. Make line c-d in same manner, using 1-inch ruler to 
make marks c-d and long ruler to draw line. 

Using 8-inch ruler, measuring from top, make line e-f. 

Using 4-inch ruler, measuring from top, make line g-h. 

Cut out corners. Compare them to each other, giving 
problems. Fold on dotted lines. 

Next day use the other sheet of manila paper and rulers 
2"4"5"10". Numerical values seen: 
2x5"=10" 10" 
5x2"=10" 2" 



12" (paper is 12" long) 



5" 

4" 

9" (paper is 9' 
wide). 



Measuring from left with 
10-inch ruler, make line a-b. 
(Fig. 2.) Cut on line. Com- 
pare the two surfaces, giving 
problems on varnishing table 
tops. (Relation 1 to 5.) 

Measuring with 4-inch ruler 
from top, make marks and draw 
line c-d. 

Paste surface e on folded 
over flaps of Fig. 1, leaving x for cover flap. 







j* 




E 




c 




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1 

• 

1 
1 
1 
1 
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MEASURING: SINGLE UNIT EULEES 



127 



LESSON XXXVI 

Book for Cuttings : Unfolded Sheets 

Rulers : 1"3"6"9". 

Materials: 10 sheets grey paper, 6"x9" ; 1 sheet grey 
or colored paper, 9"xl2"; 1 sheet manila or colored paper, 
6"x9". Fifteen inches of cord, raffia or ribbon for lacing. 

Numerical relations seen : 

3'1"=3" 2'3"=6" 

6'i"=6" 3'3"=9" 

9'1"=9" 

On each of the 10 sheets 6"x9" to be used as leaves, 
make dots A and B measuring with 1-inch ruler from the 
upper and lower left-hand corners 
(Fig. 1). 

Make dot C, using 3-inch ruler. 
Punch holes where the dots are. 
To make the cover, use the grey 
or colored paper, 12"x9" (Fig. 2). 

With the 6-inch ruler make x l °' ± 

marks A, B. Draw line and cut. Compare the two surfaces, 
giving problems on amount and cost of 
paper, linen or leather for book covers. 
(Relations 1 to 2). 

Measuring with 1-inch ruler make line 
C-D. Measuring with 3-inch ruler make 
line E-F. 
Fold left edge to meet line C-D. 
Cut on fold. Xow fold on line E-F. 





.3 a 



128 



MATHEMATICAL CONSTEUCTION 




J?iG»3- 



Using 1-inch and 3-inch rulers, as for pages, make clots 
A, B, C, and punch holes. To make decorations for cover, 
use manila paper, 6"x9". (Fig. 3.) 

Measuring with 3-inch ruler from 
left ^dge, make marks and draw line 
D-E. 

Cut on line. Compare the two sur- 
faces. Using 1-inch and 3-inch rulers, 
make dots and punch holes as before. 
This piece of paper slips under the flap of the cover. 

Use the 6-inch square which is left for any additional 
decoration desired. It can be cut into four equal oblongs. 
Bisect one of them and cut the pieces diagonally; use them 
to strengthen corners. Or, cut the 
pieces so that by making a cut J way 
down on the long edge and J way across 
on short edge, a corner is removed. Or, 
the pieces can again be bisected, making 
little squares, which can be cut in many 
ways. Another pretty decoration is the 

word "Cuttings" cut free-hand by the child, then pasted on 
the cover. 

Lace the book in any desired way. 




LESSON XXXVII 
Book With Folded Leaves for Words 



. Eulers: 6"1"7". 

Material: Six 7-inch squares of white, smooth paper; 
one 8-inch square of cover paper. 



MEASUKING 



129 



Sew together with cord 12" long. New combination seen 
6 

1 



On each of the six squares of white paper draw the line 
a-b, rising the 1-inch ruler. Cut on the line. These narrow 
A c strips can be utilized later in a weaving 

lesson. Compare them with the large sur- 
face. Now fold each of these large surfaces 
so that their long edges touch, or use a 
3-inch ruler to get the dividing line c-d. 
Measuring on this line from top and 
bottom, with 2-inch ruler make dots e, f. 
Half-way between these dots (eye judgment only) make 
another one, dot g. These are necessary only on the page 
which will be the center of the book, so the child can see 
where to place the needle. 

Fold the 8-inch cover paper into two equal oblongs. 
With the 2-inch ruler make dots as at e and /. Place 
center dot also. Sew and decorate cover in any desired way 




DEC 1 1911 



One copy del. to Cat. Div. 



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